2003

• 1.I.1B

  Algebra and Geometry

  Part IA, 2003

  (a) Write the permutation

  $(123)(234)$

  as a product of disjoint cycles. Determine its order. Compute its sign.

  (b) Elements $x$ and $y$ of a group $G$ are conjugate if there exists a $g \in G$ such that $g x g^{-1}=y .$

  Show that if permutations $x$ and $y$ are conjugate, then they have the same sign and the same order. Is the converse true? (Justify your answer with a proof or counterexample.)

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• 1.I.2D

  Algebra and Geometry

  Part IA, 2003

  Find the characteristic equation, the eigenvectors $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$, and the corresponding eigenvalues $\lambda_{\mathbf{a}}, \lambda_{\mathbf{b}}, \lambda_{\mathbf{c}}, \lambda_{\mathbf{d}}$ of the matrix

  $$A=\left(\begin{array}{cccc} i & 1 & 0 & 0 \\ 1 & i & 0 & 0 \\ 0 & 0 & i & 1 \\ 0 & 0 & -1 & i \end{array}\right)$$

  Show that $\{\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}\}$ spans the complex vector space $\mathbb{C}^{4}$.

  Consider the four subspaces of $\mathbb{C}^{4}$ defined parametrically by

  $$\mathbf{z}=s \mathbf{a}, \quad \mathbf{z}=s \mathbf{b}, \quad \mathbf{z}=s \mathbf{c}, \quad \mathbf{z}=s \mathbf{d} \quad\left(\mathbf{z} \in \mathbb{C}^{4}, s \in \mathbb{C}\right)$$

  Show that multiplication by $A$ maps three of these subspaces onto themselves, and the remaining subspace into a smaller subspace to be specified.

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• 1.II.5B

  Algebra and Geometry

  Part IA, 2003

  (a) In the standard basis of $\mathbb{R}^{2}$, write down the matrix for a rotation through an angle $\theta$ about the origin.

  (b) Let $A$ be a real $3 \times 3$ matrix such that $\operatorname{det} A=1$ and $A A^{\mathrm{T}}=I$, where $A^{\mathrm{T}}$ is the transpose of $A$.

  (i) Suppose that $A$ has an eigenvector $\mathbf{v}$ with eigenvalue 1 . Show that $A$ is a rotation through an angle $\theta$ about the line through the origin in the direction of $\mathbf{v}$, where $\cos \theta=\frac{1}{2}($ trace $A-1)$.

  (ii) Show that $A$ must have an eigenvector $\mathbf{v}$ with eigenvalue 1 .

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• 1.II.6A

  Algebra and Geometry

  Part IA, 2003

  Let $\alpha$ be a linear map

  $$\alpha: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} .$$

  Define the kernel $K$ and image $I$ of $\alpha$.

  Let $\mathbf{y} \in \mathbb{R}^{3}$. Show that the equation $\alpha \mathbf{x}=\mathbf{y}$ has a solution $\mathbf{x} \in \mathbb{R}^{3}$ if and only if $\mathbf{y} \in I$

  Let $\alpha$ have the matrix

  $$\left(\begin{array}{ccc} 1 & 1 & t \\ 0 & t & -2 b \\ 1 & t & 0 \end{array}\right)$$

  with respect to the standard basis, where $b \in \mathbb{R}$ and $t$ is a real variable. Find $K$ and $I$ for $\alpha$. Hence, or by evaluating the determinant, show that if $0<b<2$ and $\mathbf{y} \in I$ then the equation $\alpha \mathbf{x}=\mathbf{y}$ has a unique solution $\mathbf{x} \in \mathbb{R}^{3}$ for all values of $t$.

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• 1.II.7B

  Algebra and Geometry

  Part IA, 2003

  (i) State the orbit-stabilizer theorem for a group $G$ acting on a set $X$.

  (ii) Denote the group of all symmetries of the cube by $G$. Using the orbit-stabilizer theorem, show that $G$ has 48 elements.

  Does $G$ have any non-trivial normal subgroups?

  Let $L$ denote the line between two diagonally opposite vertices of the cube, and let

  $$H=\{g \in G \mid g L=L\}$$

  be the subgroup of symmetries that preserve the line. Show that $H$ is isomorphic to the direct product $S_{3} \times C_{2}$, where $S_{3}$ is the symmetric group on 3 letters and $C_{2}$ is the cyclic group of order 2 .

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• 1.II.8D

  Algebra and Geometry

  Part IA, 2003

  Let $\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ and $\mathbf{y}=\left(y_{1}, y_{2}, \ldots, y_{n}\right)$ be non-zero vectors in $\mathbb{R}^{n}$. What is meant by saying that $\mathbf{x}$ and $\mathbf{y}$ are linearly independent? What is the dimension of the subspace of $\mathbb{R}^{n}$ spanned by $\mathbf{x}$ and $\mathbf{y}$ if they are (1) linearly independent, (2) linearly dependent?

  Define the scalar product $\mathbf{x} \cdot \mathbf{y}$ for $\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}$. Define the corresponding norm $\|\mathbf{x}\|$ of $\mathbf{x} \in \mathbb{R}^{n}$. State and prove the Cauchy-Schwarz inequality, and deduce the triangle inequality.

  By means of a sketch, give a geometric interpretation of the scalar product $\mathbf{x} \cdot \mathbf{y}$ in the case $n=3$, relating the value of $\mathbf{x} \cdot \mathbf{y}$ to the angle $\alpha$ between the directions of $\mathbf{x}$ and $\mathbf{y}$.

  What is a unit vector? Let $\mathbf{u}, \mathbf{v}, \mathbf{w}$ be unit vectors in $\mathbb{R}^{3}$. Let

  $$S=\mathbf{u} \cdot \mathbf{v}+\mathbf{v} \cdot \mathbf{w}+\mathbf{w} \cdot \mathbf{u}$$

  Show that

  (i) for any fixed, linearly independent $\mathbf{u}$ and $\mathbf{v}$, the minimum of $S$ over $\mathbf{w}$ is attained when $\mathbf{w}=\lambda(\mathbf{u}+\mathbf{v})$ for some $\lambda \in \mathbb{R}$;

  (ii) $\lambda \leqslant-\frac{1}{2}$ in all cases;

  (iii) $\lambda=-1$ and $S=-3 / 2$ in the case where $\mathbf{u} \cdot \mathbf{v}=\cos (2 \pi / 3)$.

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• 3.I.1A

  Algebra and Geometry

  Part IA, 2003

  The mapping $\alpha$ of $\mathbb{R}^{3}$ into itself is a reflection in the plane $x_{2}=x_{3}$. Find the matrix $A$ of $\alpha$ with respect to any basis of your choice, which should be specified.

  The mapping $\beta$ of $\mathbb{R}^{3}$ into itself is a rotation about the line $x_{1}=x_{2}=x_{3}$ through $2 \pi / 3$, followed by a dilatation by a factor of 2 . Find the matrix $B$ of $\beta$ with respect to a choice of basis that should again be specified.

  Show explicitly that

  $$B^{3}=8 A^{2}$$

  and explain why this must hold, irrespective of your choices of bases.

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• 3.I.2B

  Algebra and Geometry

  Part IA, 2003

  Show that if a group $G$ contains a normal subgroup of order 3, and a normal subgroup of order 5 , then $G$ contains an element of order 15 .

  Give an example of a group of order 10 with no element of order $10 .$

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• 3.II.7B

  Algebra and Geometry

  Part IA, 2003

  Let $G$ be the group of Möbius transformations of $\mathbb{C} \cup\{\infty\}$ and let $X=\{\alpha, \beta, \gamma\}$ be a set of three distinct points in $\mathbb{C} \cup\{\infty\}$.

  (i) Show that there exists a $g \in G$ sending $\alpha$ to $0, \beta$ to 1 , and $\gamma$ to $\infty$.

  (ii) Hence show that if $H=\{g \in G \mid g X=X\}$, then $H$ is isomorphic to $S_{3}$, the symmetric group on 3 letters.

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• 3.II.8B

  Algebra and Geometry

  Part IA, 2003

  (a) Determine the characteristic polynomial and the eigenvectors of the matrix

  $$\left(\begin{array}{ccc} 0 & 1 & 0 \\ -1 & 2 & 0 \\ 1 & -1 & 2 \end{array}\right)$$

  Is it diagonalizable?

  (b) Show that an $n \times n$ matrix $A$ with characteristic polynomial $f(t)=(t-\mu)^{n}$ is diagonalizable if and only if $A=\mu I$.

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• 3.II.5E

  Algebra and Geometry

  Part IA, 2003

  (a) Show, using vector methods, that the distances from the centroid of a tetrahedron to the centres of opposite pairs of edges are equal. If the three distances are $u, v, w$ and if $a, b, c, d$ are the distances from the centroid to the vertices, show that

  $$u^{2}+v^{2}+w^{2}=\frac{1}{4}\left(a^{2}+b^{2}+c^{2}+d^{2}\right) .$$

  [The centroid of $k$ points in $\mathbb{R}^{3}$ with position vectors $\mathbf{x}_{i}$ is the point with position vector $\left.\frac{1}{k} \sum \mathbf{x}_{i} .\right]$

  (b) Show that

  $$|\mathbf{x}-\mathbf{a}|^{2} \cos ^{2} \alpha=[(\mathbf{x}-\mathbf{a}) \cdot \mathbf{n}]^{2},$$

  with $\mathbf{n}^{2}=1$, is the equation of a right circular double cone whose vertex has position vector a, axis of symmetry $\mathbf{n}$ and opening angle $\alpha$.

  Two such double cones, with vertices $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$, have parallel axes and the same opening angle. Show that if $\mathbf{b}=\mathbf{a}_{1}-\mathbf{a}_{2} \neq \mathbf{0}$, then the intersection of the cones lies in a plane with unit normal

  $$\mathbf{N}=\frac{\mathbf{b} \cos ^{2} \alpha-\mathbf{n}(\mathbf{n} \cdot \mathbf{b})}{\sqrt{\mathbf{b}^{2} \cos ^{4} \alpha+(\mathbf{b} \cdot \mathbf{n})^{2}\left(1-2 \cos ^{2} \alpha\right)}}$$

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• 3.II.6E

  Algebra and Geometry

  Part IA, 2003

  Derive an expression for the triple scalar product $\left(\mathbf{e}_{1} \times \mathbf{e}_{2}\right) \cdot \mathbf{e}_{3}$ in terms of the determinant of the matrix $E$ whose rows are given by the components of the three vectors $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$.

  Use the geometrical interpretation of the cross product to show that $\mathbf{e}_{a}, a=1,2,3$, will be a not necessarily orthogonal basis for $\mathbb{R}^{3}$ as long as $\operatorname{det} E \neq 0$.

  The rows of another matrix $\hat{E}$ are given by the components of three other vectors $\hat{\mathbf{e}}_{b}, b=1,2,3$. By considering the matrix $E \hat{E}^{\mathrm{T}}$, where ${ }^{\mathrm{T}}$ denotes the transpose, show that there is a unique choice of $\hat{E}$ such that $\hat{\mathbf{e}}_{b}$ is also a basis and

  $$\mathbf{e}_{a} \cdot \hat{\mathbf{e}}_{b}=\delta_{a b}$$

  Show that the new basis is given by

  $$\hat{\mathbf{e}}_{1}=\frac{\mathbf{e}_{2} \times \mathbf{e}_{3}}{\left(\mathbf{e}_{1} \times \mathbf{e}_{2}\right) \cdot \mathbf{e}_{3}} \quad \text { etc. }$$

  Show that if either $\mathbf{e}_{a}$ or $\hat{\mathbf{e}}_{b}$ is an orthonormal basis then $E$ is a rotation matrix.

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• 1.I.3B

  Analysis

  Part IA, 2003

  Define what it means for a function of a real variable to be differentiable at $x \in \mathbb{R}$.

  Prove that if a function is differentiable at $x \in \mathbb{R}$, then it is continuous there.

  Show directly from the definition that the function

  $$f(x)= \begin{cases}x^{2} \sin (1 / x) & x \neq 0 \\ 0 & x=0\end{cases}$$

  is differentiable at 0 with derivative 0 .

  Show that the derivative $f^{\prime}(x)$ is not continuous at 0 .

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• 1.I.4C

  Analysis

  Part IA, 2003

  Explain what is meant by the radius of convergence of a power series.

  Find the radius of convergence $R$ of each of the following power series: (i) $\sum_{n=1}^{\infty} n^{-2} z^{n}$, (ii) $\sum_{n=1}^{\infty}\left(n+\frac{1}{2^{n}}\right) z^{n}$.

  In each case, determine whether the series converges on the circle $|z|=R$.

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• 1.II.9F

  Analysis

  Part IA, 2003

  Prove the Axiom of Archimedes.

  Let $x$ be a real number in $[0,1]$, and let $m, n$ be positive integers. Show that the limit

  $$\lim _{m \rightarrow \infty}\left[\lim _{n \rightarrow \infty} \cos ^{2 n}(m ! \pi x)\right]$$

  exists, and that its value depends on whether $x$ is rational or irrational.

  [You may assume standard properties of the cosine function provided they are clearly stated.]

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• 1.II.10F

  Analysis

  Part IA, 2003

  State without proof the Integral Comparison Test for the convergence of a series $\sum_{n=1}^{\infty} a_{n}$ of non-negative terms.

  Determine for which positive real numbers $\alpha$ the series $\sum_{n=1}^{\infty} n^{-\alpha}$ converges.

  In each of the following cases determine whether the series is convergent or divergent: (i) $\sum_{n=3}^{\infty} \frac{1}{n \log n}$, (ii) $\sum_{n=3}^{\infty} \frac{1}{(n \log n)(\log \log n)^{2}}$, (iii) $\sum_{n=3}^{\infty} \frac{1}{n^{(1+1 / n) \log n}}$.

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• 1.II.11B

  Analysis

  Part IA, 2003

  Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous. Define the integral $\int_{a}^{b} f(x) d x$. (You are not asked to prove existence.)

  Suppose that $m, M$ are real numbers such that $m \leqslant f(x) \leqslant M$ for all $x \in[a, b]$. Stating clearly any properties of the integral that you require, show that

  $$m(b-a) \leqslant \int_{a}^{b} f(x) d x \leqslant M(b-a) .$$

  The function $g:[a, b] \rightarrow \mathbb{R}$ is continuous and non-negative. Show that

  $$m \int_{a}^{b} g(x) d x \leqslant \int_{a}^{b} f(x) g(x) d x \leqslant M \int_{a}^{b} g(x) d x$$

  Now let $f$ be continuous on $[0,1]$. By suitable choice of $g$ show that

  $$\lim _{n \rightarrow \infty} \int_{0}^{1 / \sqrt{n}} n f(x) e^{-n x} d x=f(0),$$

  and by making an appropriate change of variable, or otherwise, show that

  $$\lim _{n \rightarrow \infty} \int_{0}^{1} n f(x) e^{-n x} d x=f(0) .$$

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• 1.II.12C

  Analysis

  Part IA, 2003

  State carefully the formula for integration by parts for functions of a real variable.

  Let $f:(-1,1) \rightarrow \mathbb{R}$ be infinitely differentiable. Prove that for all $n \geqslant 1$ and all $t \in(-1,1)$,

  $f(t)=f(0)+f^{\prime}(0) t+\frac{1}{2 !} f^{\prime \prime}(0) t^{2}+\ldots+\frac{1}{(n-1) !} f^{(n-1)}(0) t^{n-1}+\frac{1}{(n-1) !} \int_{0}^{t} f^{(n)}(x)(t-x)^{n-1} d x .$

  By considering the function $f(x)=\log (1-x)$ at $x=1 / 2$, or otherwise, prove that the series

  $$\sum_{n=1}^{\infty} \frac{1}{n 2^{n}}$$

  converges to $\log 2$.

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• 2.I.1D

  Differential Equations

  Part IA, 2003

  Consider the equation

  $$\frac{d y}{d x}=1-y^{2} .$$

  Using small line segments, sketch the flow directions in $x \geqslant 0,-2 \leqslant y \leqslant 2$ implied by the right-hand side of $(*)$. Find the general solution (i) in $|y|<1$, (ii) in $|y|>1$.

  Sketch a solution curve in each of the three regions $y>1,|y|<1$, and $y<-1$.

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• 2.I.2D

  Differential Equations

  Part IA, 2003

  Consider the differential equation

  $$\frac{d x}{d t}+K x=0$$

  where $K$ is a positive constant. By using the approximate finite-difference formula

  $$\frac{d x_{n}}{d t}=\frac{x_{n+1}-x_{n-1}}{2 \delta t}$$

  where $\delta t$ is a positive constant, and where $x_{n}$ denotes the function $x(t)$ evaluated at $t=n \delta t$ for integer $n$, convert the differential equation to a difference equation for $x_{n}$.

  Solve both the differential equation and the difference equation for general initial conditions. Identify those solutions of the difference equation that agree with solutions of the differential equation over a finite interval $0 \leqslant t \leqslant T$ in the limit $\delta t \rightarrow 0$, and demonstrate the agreement. Demonstrate that the remaining solutions of the difference equation cannot agree with the solution of the differential equation in the same limit.

  [You may use the fact that, for bounded $|u|, \quad \lim _{\epsilon \rightarrow 0}(1+\epsilon u)^{1 / \epsilon}=e^{u}$.]

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• 2.II.5D

  Differential Equations

  Part IA, 2003

  (a) Show that if $\mu(x, y)$ is an integrating factor for an equation of the form

  $$f(x, y) d y+g(x, y) d x=0$$

  then $\partial(\mu f) / \partial x=\partial(\mu g) / \partial y$.

  Consider the equation

  $$\cot x d y-\tan y d x=0$$

  in the domain $0 \leqslant x \leqslant \frac{1}{2} \pi, \quad 0 \leqslant y \leqslant \frac{1}{2} \pi$. Using small line segments, sketch the flow directions in that domain. Show that $\sin x \cos y$ is an integrating factor for the equation. Find the general solution of the equation, and sketch the family of solutions that occupies the larger domain $-\frac{1}{2} \pi \leqslant x \leqslant \frac{1}{2} \pi,-\frac{1}{2} \pi \leqslant y \leqslant \frac{1}{2} \pi$.

  (b) The following example illustrates that the concept of integrating factor extends to higher-order equations. Multiply the equation

  $$\left[y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}\right] \cos ^{2} x=1$$

  by $\sec ^{2} x$, and show that the result takes the form $\frac{d}{d x} h(x, y)=0$, for some function $h(x, y)$ to be determined. Find a particular solution $y=y(x)$ such that $y(0)=0$ with $d y / d x$ finite at $x=0$, and sketch its graph in $0 \leqslant x<\frac{1}{2} \pi$.

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• $2 . \mathrm{II} . 6 \mathrm{D} \quad$

  Differential Equations

  Part IA, 2003

  Define the Wronskian $W(x)$ associated with solutions of the equation

  $$\frac{d^{2} y}{d x^{2}}+p(x) \frac{d y}{d x}+q(x) y=0$$

  and show that

  $$W(x) \propto \exp \left(-\int^{x} p(\xi) d \xi\right) .$$

  Evaluate the expression on the right when $p(x)=-2 / x$.

  Given that $p(x)=-2 / x$ and that $q(x)=-1$, show that solutions in the form of power series,

  $$y=x^{\lambda} \sum_{n=0}^{\infty} a_{n} x^{n} \quad\left(a_{0} \neq 0\right)$$

  can be found if and only if $\lambda=0$ or 3 . By constructing and solving the appropriate recurrence relations, find the coefficients $a_{n}$ for each power series.

  You may assume that the equation is satisfied by $y=\cosh x-x \sinh x$ and by $y=\sinh x-x \cosh x$. Verify that these two solutions agree with the two power series found previously, and that they give the $W(x)$ found previously, up to multiplicative constants.

  [Hint: $\left.\cosh x=1+\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}+\ldots, \quad \sinh x=x+\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}+\ldots .\right]$

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• 2.II.7D

  Differential Equations

  Part IA, 2003

  Consider the linear system

  $$\dot{\mathbf{x}}(t)-A \mathbf{x}(t)=\mathbf{z}(t)$$

  where the $n$-vector $\mathbf{z}(t)$ and the $n \times n$ matrix $A$ are given; $A$ has constant real entries, and has $n$ distinct eigenvalues $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$ and $n$ linearly independent eigenvectors $\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{n}$. Find the complementary function. Given a particular integral $\mathbf{x}_{\mathbf{p}}(t)$, write down the general solution. In the case $n=2$ show that the complementary function is purely oscillatory, with no growth or decay, if and only if

  $$\operatorname{trace} A=0 \quad \text { and } \quad \operatorname{det} A>0 .$$

  Consider the same case $n=2$ with trace $A=0$ and $\operatorname{det} A>0$ and with

  $$\mathbf{z}(t)=\mathbf{a}_{1} \exp \left(i \omega_{1} t\right)+\mathbf{a}_{2} \exp \left(i \omega_{2} t\right)$$

  where $\omega_{1}, \omega_{2}$ are given real constants. Find a particular integral when

  (i) $i \omega_{1} \neq \lambda_{1}$ and $i \omega_{2} \neq \lambda_{2}$;

  (ii) $i \omega_{1} \neq \lambda_{1}$ but $i \omega_{2}=\lambda_{2}$.

  In the case

  $$A=\left(\begin{array}{cc} 1 & 2 \\ -5 & -1 \end{array}\right)$$

  with $\mathbf{z}(t)=\left(\begin{array}{c}2 \\ 3 i-1\end{array}\right) \exp (3 i t)$, find the solution subject to the initial condition $\mathbf{x}=\left(\begin{array}{l}1 \\ 0\end{array}\right)$ at $t=0$.

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• 2.II.8D

  Differential Equations

  Part IA, 2003

  For all solutions of

  $$\begin{aligned} &\dot{x}=\frac{1}{2} \alpha x+y-2 y^{3} \\ &\dot{y}=-x \end{aligned}$$

  show that $d K / d t=\alpha x^{2}$ where

  $$K=K(x, y)=x^{2}+y^{2}-y^{4}$$

  In the case $\alpha=0$, analyse the properties of the critical points and sketch the phase portrait, including the special contours for which $K(x, y)=\frac{1}{4}$. Comment on the asymptotic behaviour, as $t \rightarrow \infty$, of solution trajectories that pass near each critical point, indicating whether or not any such solution trajectories approach from, or recede to, infinity.

  Briefly discuss how the picture changes when $\alpha$ is made small and positive, using your result for $d K / d t$ to describe, in qualitative terms, how solution trajectories cross $K$-contours.

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• 4.I.3E

  Dynamics

  Part IA, 2003

  Because of an accident on launching, a rocket of unladen mass $M$ lies horizontally on the ground. It initially contains fuel of mass $m_{0}$, which ignites and is emitted horizontally at a constant rate and at uniform speed $u$ relative to the rocket. The rocket is initially at rest. If the coefficient of friction between the rocket and the ground is $\mu$, and the fuel is completely burnt in a total time $T$, show that the final speed of the rocket is

  $$u \log \left(\frac{M+m_{0}}{M}\right)-\mu g T$$

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• 4.I.4E

  Dynamics

  Part IA, 2003

  Write down an expression for the total momentum $\mathbf{P}$ and angular momentum $\mathbf{L}$ with respect to an origin $O$ of a system of $n$ point particles of masses $m_{i}$, position vectors (with respect to $O) \mathbf{x}_{i}$, and velocities $\mathbf{v}_{i}, i=1, \ldots, n$.

  Show that with respect to a new origin $O^{\prime}$ the total momentum $\mathbf{P}^{\prime}$ and total angular momentum $\mathbf{L}^{\prime}$ are given by

  $$\mathbf{P}^{\prime}=\mathbf{P}, \quad \mathbf{L}^{\prime}=\mathbf{L}-\mathbf{b} \times \mathbf{P}$$

  and hence

  $$\mathbf{L}^{\prime} \cdot \mathbf{P}^{\prime}=\mathbf{L} \cdot \mathbf{P},$$

  where $\mathbf{b}$ is the constant vector displacement of $O^{\prime}$ with respect to $O$. How does $\mathbf{L} \times \mathbf{P}$ change under change of origin?

  Hence show that either

  (1) the total momentum vanishes and the total angular momentum is independent of origin, or

  (2) by choosing $\mathbf{b}$ in a way that should be specified, the total angular momentum with respect to $O^{\prime}$ can be made parallel to the total momentum.

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• 4.II.9E

  Dynamics

  Part IA, 2003

  Write down the equation of motion for a point particle with mass $m$, charge $e$, and position vector $\mathbf{x}(t)$ moving in a time-dependent magnetic field $\mathbf{B}(\mathbf{x}, t)$ with vanishing electric field, and show that the kinetic energy of the particle is constant. If the magnetic field is constant in direction, show that the component of velocity in the direction of $\mathbf{B}$ is constant. Show that, in general, the angular momentum of the particle is not conserved.

  Suppose that the magnetic field is independent of time and space and takes the form $\mathbf{B}=(0,0, B)$ and that $\dot{A}$ is the rate of change of area swept out by a radius vector joining the origin to the projection of the particle's path on the $(x, y)$ plane. Obtain the equation

  $$\frac{d}{d t}\left(m \dot{A}+\frac{e B r^{2}}{4}\right)=0,$$

  where $(r, \theta)$ are plane polar coordinates. Hence obtain an equation replacing the equation of conservation of angular momentum.

  Show further, using energy conservation and $(*)$, that the equations of motion in plane polar coordinates may be reduced to the first order non-linear system

  $$\begin{gathered} \dot{r}=\sqrt{v^{2}-\left(\frac{2 c}{m r}-\frac{e r B}{2 m}\right)^{2}} \\ \dot{\theta}=\frac{2 c}{m r^{2}}-\frac{e B}{2 m} \end{gathered}$$

  where $v$ and $c$ are constants.

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• 4.II.10E

  Dynamics

  Part IA, 2003

  Write down the equations of motion for a system of $n$ gravitating particles with masses $m_{i}$, and position vectors $\mathbf{x}_{i}, i=1,2, \ldots, n$.

  The particles undergo a motion for which $\mathbf{x}_{i}(t)=a(t) \mathbf{a}_{i}$, where the vectors $\mathbf{a}_{i}$ are independent of time $t$. Show that the equations of motion will be satisfied as long as the function $a(t)$ satisfies

  $$\ddot{a}=-\frac{\Lambda}{a^{2}},$$

  where $\Lambda$ is a constant and the vectors $\mathbf{a}_{i}$ satisfy

  $$\Lambda m_{i} \mathbf{a}_{i}=\mathbf{G}_{i}=\sum_{j \neq i} \frac{G m_{i} m_{j}\left(\mathbf{a}_{i}-\mathbf{a}_{j}\right)}{\left|\mathbf{a}_{i}-\mathbf{a}_{j}\right|^{3}}$$

  Show that $(*)$ has as first integral

  $$\frac{\dot{a}^{2}}{2}-\frac{\Lambda}{a}=\frac{k}{2}$$

  where $k$ is another constant. Show that

  $$\mathbf{G}_{i}=\nabla_{i} W$$

  where $\boldsymbol{\nabla}_{i}$ is the gradient operator with respect to $\mathbf{a}_{i}$ and

  $$W=-\sum_{i} \sum_{j<i} \frac{G m_{i} m_{j}}{\left|\mathbf{a}_{i}-\mathbf{a}_{j}\right|} .$$

  Using Euler's theorem for homogeneous functions (see below), or otherwise, deduce that

  $$\sum_{i} \mathbf{a}_{i} \cdot \mathbf{G}_{i}=-W .$$

  Hence show that all solutions of $(* *)$ satisfy

  $$\Lambda I=-W$$

  where

  $$I=\sum_{i} m_{i} \mathbf{a}_{i}^{2}$$

  Deduce that $\Lambda$ must be positive and that the total kinetic energy plus potential energy of the system of particles is equal to $\frac{k}{2} I$.

  [Euler's theorem states that if

  $$f(\lambda x, \lambda y, \lambda z, \ldots)=\lambda^{p} f(x, y, z, \ldots)$$

  then

  $$\left.x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}+z \frac{\partial f}{\partial z}+\ldots=p f .\right]$$

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• 4.II.11E

  Dynamics

  Part IA, 2003

  State the parallel axis theorem and use it to calculate the moment of inertia of a uniform hemisphere of mass $m$ and radius $a$ about an axis through its centre of mass and parallel to the base.

  [You may assume that the centre of mass is located at a distance $\frac{3}{8}$ a from the flat face of the hemisphere, and that the moment of inertia of a full sphere about its centre is $\frac{2}{5} M a^{2}$, with $M=2 m$.]

  The hemisphere initially rests on a rough horizontal plane with its base vertical. It is then released from rest and subsequently rolls on the plane without slipping. Let $\theta$ be the angle that the base makes with the horizontal at time $t$. Express the instantaneous speed of the centre of mass in terms of $b$ and the rate of change of $\theta$, where $b$ is the instantaneous distance from the centre of mass to the point of contact with the plane. Hence write down expressions for the kinetic energy and potential energy of the hemisphere and deduce that

  $$\left(\frac{d \theta}{d t}\right)^{2}=\frac{15 g \cos \theta}{(28-15 \cos \theta) a}$$

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• 4.II.12E

  Dynamics

  Part IA, 2003

  Let $(r, \theta)$ be plane polar coordinates and $\mathbf{e}_{r}$ and $\mathbf{e}_{\theta}$ unit vectors in the direction of increasing $r$ and $\theta$ respectively. Show that the velocity of a particle moving in the plane with polar coordinates $(r(t), \theta(t))$ is given by

  $$\dot{\mathbf{x}}=\dot{r} \mathbf{e}_{r}+r \dot{\theta} \mathbf{e}_{\theta},$$

  and that the unit normal $\mathbf{n}$ to the particle path is parallel to

  $$r \dot{\theta} \mathbf{e}_{r}-\dot{r} \mathbf{e}_{\theta} \text {. }$$

  Deduce that the perpendicular distance $p$ from the origin to the tangent of the curve $r=r(\theta)$ is given by

  $$\frac{r^{2}}{p^{2}}=1+\frac{1}{r^{2}}\left(\frac{d r}{d \theta}\right)^{2}$$

  The particle, whose mass is $m$, moves under the influence of a central force with potential $V(r)$. Use the conservation of energy $E$ and angular momentum $h$ to obtain the equation

  $$\frac{1}{p^{2}}=\frac{2 m(E-V(r))}{h^{2}}$$

  Hence express $\theta$ as a function of $r$ as the integral

  $$\theta=\int \frac{h r^{-2} d r}{\sqrt{2 m\left(E-V_{\mathrm{eff}}(r)\right)}}$$

  where

  $$V_{\mathrm{eff}}(r)=V(r)+\frac{h^{2}}{2 m r^{2}}$$

  Evaluate the integral and describe the orbit when $V(r)=\frac{c}{r^{2}}$, with $c$ a positive constant.

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• 4.I.1C

  Numbers and Sets

  Part IA, 2003

  (i) Prove by induction or otherwise that for every $n \geqslant 1$,

  $$\sum_{r=1}^{n} r^{3}=\left(\sum_{r=1}^{n} r\right)^{2}$$

  (ii) Show that the sum of the first $n$ positive cubes is divisible by 4 if and only if $n \equiv 0$ or $3(\bmod 4)$.

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• 4.I.2C

  Numbers and Sets

  Part IA, 2003

  What is an equivalence relation? For each of the following pairs $(X, \sim)$, determine whether or not $\sim$ is an equivalence relation on $X$ :

  (i) $X=\mathbb{R}, x \sim y$ iff $x-y$ is an even integer;

  (ii) $X=\mathbb{C} \backslash\{0\}, x \sim y$ iff $x \bar{y} \in \mathbb{R}$;

  (iii) $X=\mathbb{C} \backslash\{0\}, x \sim y$ iff $x \bar{y} \in \mathbb{Z}$;

  (iv) $X=\mathbb{Z} \backslash\{0\}, x \sim y$ iff $x^{2}-y^{2}$ is $\pm 1$ times a perfect square.

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• 4.II.5C

  Numbers and Sets

  Part IA, 2003

  Define what is meant by the term countable. Show directly from your definition that if $X$ is countable, then so is any subset of $X$.

  Show that $\mathbb{N} \times \mathbb{N}$ is countable. Hence or otherwise, show that a countable union of countable sets is countable. Show also that for any $n \geqslant 1, \mathbb{N}^{n}$ is countable.

  A function $f: \mathbb{Z} \rightarrow \mathbb{N}$ is periodic if there exists a positive integer $m$ such that, for every $x \in \mathbb{Z}, f(x+m)=f(x)$. Show that the set of periodic functions $f: \mathbb{Z} \rightarrow \mathbb{N}$ is countable.

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• 4.II.6C

  Numbers and Sets

  Part IA, 2003

  (i) Prove Wilson's theorem: if $p$ is prime then $(p-1) ! \equiv-1(\bmod p)$.

  Deduce that if $p \equiv 1(\bmod 4)$ then

  $$\left(\left(\frac{p-1}{2}\right) !\right)^{2} \equiv-1 \quad(\bmod p)$$

  (ii) Suppose that $p$ is a prime of the form $4 k+3$. Show that if $x^{4} \equiv 1(\bmod p)$ then $x^{2} \equiv 1(\bmod p)$.

  (iii) Deduce that if $p$ is an odd prime, then the congruence

  $$x^{2} \equiv-1 \quad(\bmod p)$$

  has exactly two solutions ( $\operatorname{modulo} p)$ if $p \equiv 1(\bmod 4)$, and none otherwise.

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• 4.II.7C

  Numbers and Sets

  Part IA, 2003

  Let $m, n$ be integers. Explain what is their greatest common divisor $(m, n)$. Show from your definition that, for any integer $k, \quad(m, n)=(m+k n, n)$.

  State Bezout's theorem, and use it to show that if $p$ is prime and $p$ divides $m n$, then $p$ divides at least one of $m$ and $n$.

  The Fibonacci sequence $0,1,1,2,3,5,8, \ldots$ is defined by $x_{0}=0, x_{1}=1$ and $x_{n+1}=x_{n}+x_{n-1}$ for $n \geqslant 1$. Prove:

  (i) $\left(x_{n+1}, x_{n}\right)=1$ and $\left(x_{n+2}, x_{n}\right)=1$ for every $n \geqslant 0$;

  (ii) $x_{n+3} \equiv x_{n}(\bmod 2)$ and $x_{n+8} \equiv x_{n}(\bmod 3)$ for every $n \geqslant 0$;

  (iii) if $n \equiv 0(\bmod 5)$ then $x_{n} \equiv 0(\bmod 5)$.

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• 4.II.8C

  Numbers and Sets

  Part IA, 2003

  Let $X$ be a finite set with $n$ elements. How many functions are there from $X$ to $X$ ? How many relations are there on $X$ ?

  Show that the number of relations $R$ on $X$ such that, for each $y \in X$, there exists at least one $x \in X$ with $x R y$, is $\left(2^{n}-1\right)^{n}$.

  Using the inclusion-exclusion principle or otherwise, deduce that the number of such relations $R$ for which, in addition, for each $x \in X$, there exists at least one $y \in X$ with $x R y$, is

  $$\sum_{k=0}^{n}(-1)^{k}\left(\begin{array}{l} n \\ k \end{array}\right)\left(2^{n-k}-1\right)^{n}$$

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• 2.I.3F

  Probability

  Part IA, 2003

  (a) Define the probability generating function of a random variable. Calculate the probability generating function of a binomial random variable with parameters $n$ and $p$, and use it to find the mean and variance of the random variable.

  (b) $X$ is a binomial random variable with parameters $n$ and $p, Y$ is a binomial random variable with parameters $m$ and $p$, and $X$ and $Y$ are independent. Find the distribution of $X+Y$; that is, determine $P\{X+Y=k\}$ for all possible values of $k$.

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• 2.I.4F

  Probability

  Part IA, 2003

  The random variable $X$ is uniformly distributed on the interval $[0,1]$. Find the distribution function and the probability density function of $Y$, where

  $$Y=\frac{3 X}{1-X}$$

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• 2.II.9F

  Probability

  Part IA, 2003

  State the inclusion-exclusion formula for the probability that at least one of the events $A_{1}, A_{2}, \ldots, A_{n}$ occurs.

  After a party the $n$ guests take coats randomly from a pile of their $n$ coats. Calculate the probability that no-one goes home with the correct coat.

  Let $p(m, n)$ be the probability that exactly $m$ guests go home with the correct coats. By relating $p(m, n)$ to $p(0, n-m)$, or otherwise, determine $p(m, n)$ and deduce that

  $$\lim _{n \rightarrow \infty} p(m, n)=\frac{1}{e m !} .$$

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• 2.II.10F

  Probability

  Part IA, 2003

  The random variables $X$ and $Y$ each take values in $\{0,1\}$, and their joint distribution $p(x, y)=P\{X=x, Y=y\}$ is given by

  $$p(0,0)=a, \quad p(0,1)=b, \quad p(1,0)=c, \quad p(1,1)=d .$$

  Find necessary and sufficient conditions for $X$ and $Y$ to be (i) uncorrelated; (ii) independent.

  Are the conditions established in (i) and (ii) equivalent?

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• 2.II.11F

  Probability

  Part IA, 2003

  A laboratory keeps a population of aphids. The probability of an aphid passing a day uneventfully is $q<1$. Given that a day is not uneventful, there is probability $r$ that the aphid will have one offspring, probability $s$ that it will have two offspring and probability $t$ that it will die, where $r+s+t=1$. Offspring are ready to reproduce the next day. The fates of different aphids are independent, as are the events of different days. The laboratory starts out with one aphid.

  Let $X_{1}$ be the number of aphids at the end of the first day. What is the expected value of $X_{1}$ ? Determine an expression for the probability generating function of $X_{1}$.

  Show that the probability of extinction does not depend on $q$, and that if $2 r+3 s \leqslant 1$ then the aphids will certainly die out. Find the probability of extinction if $r=1 / 5, s=2 / 5$ and $t=2 / 5$.

  [Standard results on branching processes may be used without proof, provided that they are clearly stated.]

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• 2.II.12F

  Probability

  Part IA, 2003

  Planet Zog is a ball with centre $O$. Three spaceships $A, B$ and $C$ land at random on its surface, their positions being independent and each uniformly distributed on its surface. Calculate the probability density function of the angle $\angle A O B$ formed by the lines $O A$ and $O B$.

  Spaceships $A$ and $B$ can communicate directly by radio if $\angle A O B<\pi / 2$, and similarly for spaceships $B$ and $C$ and spaceships $A$ and $C$. Given angle $\angle A O B=\gamma<\pi / 2$, calculate the probability that $C$ can communicate directly with either $A$ or $B$. Given angle $\angle A O B=\gamma>\pi / 2$, calculate the probability that $C$ can communicate directly with both $A$ and $B$. Hence, or otherwise, show that the probability that all three spaceships can keep in in touch (with, for example, $A$ communicating with $B$ via $C$ if necessary) is $(\pi+2) /(4 \pi)$.

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• 3.I.3A

  Vector Calculus

  Part IA, 2003

  Sketch the curve $y^{2}=x^{2}+1$. By finding a parametric representation, or otherwise, determine the points on the curve where the radius of curvature is least, and compute its value there.

  [Hint: you may use the fact that the radius of curvature of a parametrized curve $(x(t), y(t))$ is $\left(\dot{x}^{2}+\dot{y}^{2}\right)^{3 / 2} /|\dot{x} \ddot{y}-\ddot{x} \dot{y}|$.]

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• 3.I.4A

  Vector Calculus

  Part IA, 2003

  Suppose $V$ is a region in $\mathbb{R}^{3}$, bounded by a piecewise smooth closed surface $S$, and $\phi(\mathbf{x})$ is a scalar field satisfying

  $$\begin{aligned} \nabla^{2} \phi &=0 \text { in } V, \\ \text { and } \phi &=f(\mathbf{x}) \text { on } S . \end{aligned}$$

  Prove that $\phi$ is determined uniquely in $V$.

  How does the situation change if the normal derivative of $\phi$ rather than $\phi$ itself is specified on $S$ ?

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• 3.II.9A

  Vector Calculus

  Part IA, 2003

  Let $C$ be the closed curve that is the boundary of the triangle $T$ with vertices at the points $(1,0,0),(0,1,0)$ and $(0,0,1)$.

  Specify a direction along $C$ and consider the integral

  $$\oint_{C} \mathbf{A} \cdot d \mathbf{x}$$

  where $\mathbf{A}=\left(z^{2}-y^{2}, x^{2}-z^{2}, y^{2}-x^{2}\right)$. Explain why the contribution to the integral is the same from each edge of $C$, and evaluate the integral.

  State Stokes's theorem and use it to evaluate the surface integral

  $$\int_{T}(\boldsymbol{\nabla} \times \mathbf{A}) \cdot d \mathbf{S},$$

  the components of the normal to $T$ being positive.

  Show that $d \mathbf{S}$ in the above surface integral can be written in the form $(1,1,1) d y d z$.

  Use this to verify your result by a direct calculation of the surface integral.

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• 3.II.10A

  Vector Calculus

  Part IA, 2003

  Write down an expression for the Jacobian $J$ of a transformation

  $$(x, y, z) \rightarrow(u, v, w)$$

  Use it to show that

  $$\int_{D} f d x d y d z=\int_{\Delta} \phi|J| d u d v d w$$

  where $D$ is mapped one-to-one onto $\Delta$, and

  $$\phi(u, v, w)=f(x(u, v, w), y(u, v, w), z(u, v, w))$$

  Find a transformation that maps the ellipsoid $D$,

  $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} \leqslant 1$$

  onto a sphere. Hence evaluate

  $$\int_{D} x^{2} d x d y d z$$

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• 3.II.11A

  Vector Calculus

  Part IA, 2003

  (a) Prove the identity

  $$\boldsymbol{\nabla}(\mathbf{F} \cdot \mathbf{G})=(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{G}+(\mathbf{G} \cdot \boldsymbol{\nabla}) \mathbf{F}+\mathbf{F} \times(\boldsymbol{\nabla} \times \mathbf{G})+\mathbf{G} \times(\boldsymbol{\nabla} \times \mathbf{F})$$

  (b) If $\mathbf{E}$ is an irrotational vector field $(\boldsymbol{\nabla} \times \mathbf{E}=\mathbf{0}$ everywhere $)$, prove that there exists a scalar potential $\phi(\mathbf{x})$ such that $\mathbf{E}=-\boldsymbol{\nabla} \phi$.

  Show that

  $$\left(2 x y^{2} z e^{-x^{2} z},-2 y e^{-x^{2} z}, x^{2} y^{2} e^{-x^{2} z}\right)$$

  is irrotational, and determine the corresponding potential $\phi$.

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• 3.II.12A

  Vector Calculus

  Part IA, 2003

  State the divergence theorem. By applying this to $f(\mathbf{x}) \mathbf{k}$, where $f(\mathbf{x})$ is a scalar field in a closed region $V$ in $\mathbb{R}^{3}$ bounded by a piecewise smooth surface $S$, and $\mathbf{k}$ an arbitrary constant vector, show that

  $$\int_{V} \nabla f d V=\int_{S} f d \mathbf{S}$$

  A vector field $\mathbf{G}$ satisfies

  $$\begin{gathered} \boldsymbol{\nabla} \cdot \mathbf{G}=\rho(\mathbf{x}) \\ \text { with } \quad \rho(\mathbf{x})= \begin{cases}\rho_{0} & |\mathbf{x}| \leqslant a \\ 0 & |\mathbf{x}|>a\end{cases} \end{gathered}$$

  By applying the divergence theorem to $\int_{V} \nabla \cdot \mathbf{G} d V$, prove Gauss's law

  $$\int_{S} \mathbf{G} \cdot d \mathbf{S}=\int_{V} \rho(\mathbf{x}) d V$$

  where $S$ is the piecewise smooth surface bounding the volume $V$.

  Consider the spherically symmetric solution

  $$\mathbf{G}(\mathbf{x})=G(r) \frac{\mathbf{x}}{r}$$

  where $r=|\mathbf{x}|$. By using Gauss's law with $S$ a sphere of radius $r$, centre $\mathbf{0}$, in the two cases $0<r \leqslant a$ and $r>a$, show that

  $$\mathbf{G}(\mathbf{x})= \begin{cases}\frac{\rho_{0}}{3} \mathbf{x} & r \leqslant a \\ \frac{\rho_{0}}{3}\left(\frac{a}{r}\right)^{3} \mathbf{x} & r>a\end{cases}$$

  The scalar field $f(\mathbf{x})$ satisfies $\mathbf{G}=\nabla f$. Assuming that $f \rightarrow 0$ as $r \rightarrow \infty$, and that $f$ is continuous at $r=a$, find $f$ everywhere.

  By using a symmetry argument, explain why $(*)$ is clearly satisfied for this $f$ if $S$ is any sphere centred at the origin.

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• 1.I.1F

  Analysis II

  Part IB, 2003

  Let $E$ be a subset of $\mathbb{R}^{n}$. Prove that the following conditions on $E$ are equivalent:

  (i) $E$ is closed and bounded.

  (ii) $E$ has the Bolzano-Weierstrass property (i.e., every sequence in $E$ has a subsequence convergent to a point of $E$ ).

  (iii) Every continuous real-valued function on $E$ is bounded.

  [The Bolzano-Weierstrass property for bounded closed intervals in $\mathbb{R}^{1}$ may be assumed.]

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• 1.II.10F

  Analysis II

  Part IB, 2003

  Explain briefly what is meant by a metric space, and by a Cauchy sequence in a metric space.

  A function $d: X \times X \rightarrow \mathbb{R}$ is called a pseudometric on $X$ if it satisfies all the conditions for a metric except the requirement that $d(x, y)=0$ implies $x=y$. If $d$ is a pseudometric on $X$, show that the binary relation $R$ on $X$ defined by $x R y \Leftrightarrow d(x, y)=0$ is an equivalence relation, and that the function $d$ induces a metric on the set $X / R$ of equivalence classes.

  Now let $(X, d)$ be a metric space. If $\left(x_{n}\right)$ and $\left(y_{n}\right)$ are Cauchy sequences in $X$, show that the sequence whose $n$th term is $d\left(x_{n}, y_{n}\right)$ is a Cauchy sequence of real numbers. Deduce that the function $\bar{d}$ defined by

  $$\bar{d}\left(\left(x_{n}\right),\left(y_{n}\right)\right)=\lim _{n \rightarrow \infty} d\left(x_{n}, y_{n}\right)$$

  is a pseudometric on the set $C$ of all Cauchy sequences in $X$. Show also that there is an isometric embedding (that is, a distance-preserving mapping) $X \rightarrow C / R$, where $R$ is the equivalence relation on $C$ induced by the pseudometric $\bar{d}$ as in the previous paragraph. Under what conditions on $X$ is $X \rightarrow C / R$ bijective? Justify your answer.

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• 2.I.1F

  Analysis II

  Part IB, 2003

  Explain what it means for a function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{1}$ to be differentiable at a point $(a, b)$. Show that if the partial derivatives $\partial f / \partial x$ and $\partial f / \partial y$ exist in a neighbourhood of $(a, b)$ and are continuous at $(a, b)$ then $f$ is differentiable at $(a, b)$.

  Let

  $$f(x, y)=\frac{x y}{x^{2}+y^{2}} \quad((x, y) \neq(0,0))$$

  and $f(0,0)=0$. Do the partial derivatives of $f$ exist at $(0,0) ?$ Is $f$ differentiable at $(0,0) ?$ Justify your answers.

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• 2.II.10F

  Analysis II

  Part IB, 2003

  Let $V$ be the space of $n \times n$ real matrices. Show that the function

  $$N(A)=\sup \left\{\|A \mathbf{x}\|: \mathbf{x} \in \mathbb{R}^{n},\|\mathbf{x}\|=1\right\}$$

  (where $\|-\|$ denotes the usual Euclidean norm on $\mathbb{R}^{n}$ ) defines a norm on $V$. Show also that this norm satisfies $N(A B) \leqslant N(A) N(B)$ for all $A$ and $B$, and that if $N(A)<\epsilon$ then all entries of $A$ have absolute value less than $\epsilon$. Deduce that any function $f: V \rightarrow \mathbb{R}$ such that $f(A)$ is a polynomial in the entries of $A$ is continuously differentiable.

  Now let $d: V \rightarrow \mathbb{R}$ be the mapping sending a matrix to its determinant. By considering $d(I+H)$ as a polynomial in the entries of $H$, show that the derivative $d^{\prime}(I)$ is the function $H \mapsto \operatorname{tr} H$. Deduce that, for any $A, d^{\prime}(A)$ is the mapping $H \mapsto \operatorname{tr}((\operatorname{adj} A) H)$, where $\operatorname{adj} A$ is the adjugate of $A$, i.e. the matrix of its cofactors.

  [Hint: consider first the case when $A$ is invertible. You may assume the results that the set $U$ of invertible matrices is open in $V$ and that its closure is the whole of $V$, and the identity $(\operatorname{adj} A) A=\operatorname{det} A . I$.]

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• 3.I.1F

  Analysis II

  Part IB, 2003

  Let $V$ be the vector space of continuous real-valued functions on $[-1,1]$. Show that the function

  $$\|f\|=\int_{-1}^{1}|f(x)| d x$$

  defines a norm on $V$.

  Let $f_{n}(x)=x^{n}$. Show that $\left(f_{n}\right)$ is a Cauchy sequence in $V$. Is $\left(f_{n}\right)$ convergent? Justify your answer.

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• $3 . \mathrm{II} . 11 \mathrm{~F} \quad$

  Analysis II

  Part IB, 2003

  State and prove the Contraction Mapping Theorem.

  Let $(X, d)$ be a bounded metric space, and let $F$ denote the set of all continuous maps $X \rightarrow X$. Let $\rho: F \times F \rightarrow \mathbb{R}$ be the function

  $$\rho(f, g)=\sup \{d(f(x), g(x)): x \in X\}$$

  Show that $\rho$ is a metric on $F$, and that $(F, \rho)$ is complete if $(X, d)$ is complete. [You may assume that a uniform limit of continuous functions is continuous.]

  Now suppose that $(X, d)$ is complete. Let $C \subseteq F$ be the set of contraction mappings, and let $\theta: C \rightarrow X$ be the function which sends a contraction mapping to its unique fixed point. Show that $\theta$ is continuous. [Hint: fix $f \in C$ and consider $d(\theta(g), f(\theta(g)))$, where $g \in C$ is close to $f$.]

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• 4.I.1F

  Analysis II

  Part IB, 2003

  Explain what it means for a sequence of functions $\left(f_{n}\right)$ to converge uniformly to a function $f$ on an interval. If $\left(f_{n}\right)$ is a sequence of continuous functions converging uniformly to $f$ on a finite interval $[a, b]$, show that

  $$\int_{a}^{b} f_{n}(x) d x \longrightarrow \int_{a}^{b} f(x) d x \quad \text { as } n \rightarrow \infty$$

  Let $f_{n}(x)=x \exp (-x / n) / n^{2}, x \geqslant 0$. Does $f_{n} \rightarrow 0$ uniformly on $[0, \infty) ?$ Does $\int_{0}^{\infty} f_{n}(x) d x \rightarrow 0$ ? Justify your answers.

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• 4.II.10F

  Analysis II

  Part IB, 2003

  Let $\left(f_{n}\right)_{n \geqslant 1}$ be a sequence of continuous complex-valued functions defined on a set $E \subseteq \mathbb{C}$, and converging uniformly on $E$ to a function $f$. Prove that $f$ is continuous on $E$.

  State the Weierstrass $M$-test for uniform convergence of a series $\sum_{n=1}^{\infty} u_{n}(z)$ of complex-valued functions on a set $E$.

  Now let $f(z)=\sum_{n=1}^{\infty} u_{n}(z)$, where

  $$u_{n}(z)=n^{-2} \sec (\pi z / 2 n) .$$

  Prove carefully that $f$ is continuous on $\mathbb{C} \backslash \mathbb{Z}$.

  [You may assume the inequality $|\cos z| \geqslant|\cos (\operatorname{Re} z)| \cdot]$

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• 1.I.7B

  Complex Methods

  Part IB, 2003

  Let $u(x, y)$ and $v(x, y)$ be a pair of conjugate harmonic functions in a domain $D$.

  Prove that

  $$U(x, y)=e^{-2 u v} \cos \left(u^{2}-v^{2}\right) \quad \text { and } \quad V(x, y)=e^{-2 u v} \sin \left(u^{2}-v^{2}\right)$$

  also form a pair of conjugate harmonic functions in $D$.

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• 2.I.7B

  Complex Methods

  Part IB, 2003

  (a) Using the residue theorem, evaluate

  $$\int_{|z|=1}\left(z-\frac{1}{z}\right)^{2 n} \frac{d z}{z}$$

  (b) Deduce that

  $$\int_{0}^{2 \pi} \sin ^{2 n} t d t=\frac{\pi}{2^{2 n-1}} \frac{(2 n) !}{(n !)^{2}}$$

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• 2.II.16B

  Complex Methods

  Part IB, 2003

  (a) Show that if $f$ satisfies the equation

  $$f^{\prime \prime}(x)-x^{2} f(x)=\mu f(x), \quad x \in \mathbb{R},$$

  where $\mu$ is a constant, then its Fourier transform $\widehat{f}$ satisfies the same equation, i.e.

  $$\widehat{f}^{\prime \prime}(\lambda)-\lambda^{2} \widehat{f}(\lambda)=\mu \widehat{f}(\lambda) .$$

  (b) Prove that, for each $n \geq 0$, there is a polynomial $p_{n}(x)$ of degree $n$, unique up to multiplication by a constant, such that

  $$f_{n}(x)=p_{n}(x) e^{-x^{2} / 2}$$

  is a solution of $(*)$ for some $\mu=\mu_{n}$.

  (c) Using the fact that $g(x)=e^{-x^{2} / 2}$ satisfies $\widehat{g}=c g$ for some constant $c$, show that the Fourier transform of $f_{n}$ has the form

  $$\widehat{f_{n}}(\lambda)=q_{n}(\lambda) e^{-\lambda^{2} / 2}$$

  where $q_{n}$ is also a polynomial of degree $n$.

  (d) Deduce that the $f_{n}$ are eigenfunctions of the Fourier transform operator, i.e. $\widehat{f_{n}}(x)=c_{n} f_{n}(x)$ for some constants $c_{n} .$

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• 1.II.16B

  Complex Methods

  Part IB, 2003

  Sketch the region $A$ which is the intersection of the discs

  $$D_{0}=\{z \in \mathbb{C}:|z|<1\} \quad \text { and } \quad D_{1}=\{z \in \mathbb{C}:|z-(1+i)|<1\} .$$

  Find a conformal mapping that maps $A$ onto the right half-plane $H=\{z \in \mathbb{C}: \operatorname{Re} z>0\}$. Also find a conformal mapping that maps $A$ onto $D_{0}$.

  [Hint: You may find it useful to consider maps of the form $w(z)=\frac{a z+b}{c z+d}$.]

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• 4.I.8B

  Complex Methods

  Part IB, 2003

  Find the Laurent series centred on 0 for the function

  $$f(z)=\frac{1}{(z-1)(z-2)}$$

  in each of the domains (a) $|z|<1$, (b) $1<|z|<2$, (c) $|z|>2$.

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• 4.II.17B

  Complex Methods

  Part IB, 2003

  Let

  $$f(z)=\frac{z^{m}}{1+z^{n}}, \quad n>m+1, \quad m, n \in \mathbb{N},$$

  and let $C_{R}$ be the boundary of the domain

  $$D_{R}=\left\{z=r e^{i \theta}: 0<r<R, \quad 0<\theta<\frac{2 \pi}{n}\right\}, \quad R>1 .$$

  (a) Using the residue theorem, determine

  $$\int_{C_{R}} f(z) d z$$

  (b) Show that the integral of $f(z)$ along the circular part $\gamma_{R}$ of $C_{R}$ tends to 0 as $R \rightarrow \infty$.

  (c) Deduce that

  $$\int_{0}^{\infty} \frac{x^{m}}{1+x^{n}} d x=\frac{\pi}{n \sin \frac{\pi(m+1)}{n}}$$

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• 1.I.6C

  Fluid Dynamics

  Part IB, 2003

  An unsteady fluid flow has velocity field given in Cartesian coordinates $(x, y, z)$ by $\mathbf{u}=(1, x t, 0)$, where $t$ denotes time. Dye is released into the fluid from the origin continuously. Find the position at time $t$ of the dye particle that was released at time $s$ and hence show that the dye streak lies along the curve

  $$y=\frac{1}{2} t x^{2}-\frac{1}{6} x^{3}$$

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• 1.II.15C

  Fluid Dynamics

  Part IB, 2003

  Starting from the Euler equations for incompressible, inviscid flow

  $$\rho \frac{D \mathbf{u}}{D t}=-\nabla p, \quad \nabla \cdot \mathbf{u}=0$$

  derive the vorticity equation governing the evolution of the vorticity $\boldsymbol{\omega}=\nabla \times \mathbf{u}$.

  Consider the flow

  $$\mathbf{u}=\beta(-x,-y, 2 z)+\Omega(t)(-y, x, 0)$$

  in Cartesian coordinates $(x, y, z)$, where $t$ is time and $\beta$ is a constant. Compute the vorticity and show that it evolves in time according to

  $$\boldsymbol{\omega}=\omega_{0} \mathrm{e}^{2 \beta t} \mathbf{k}$$

  where $\omega_{0}$ is the initial magnitude of the vorticity and $\mathbf{k}$ is a unit vector in the $z$-direction.

  Show that the material curve $C(t)$ that takes the form

  $$x^{2}+y^{2}=1 \quad \text { and } \quad z=1$$

  at $t=0$ is given later by

  $$x^{2}+y^{2}=a^{2}(t) \quad \text { and } \quad z=\frac{1}{a^{2}(t)},$$

  where the function $a(t)$ is to be determined.

  Calculate the circulation of $\mathbf{u}$ around $C$ and state how this illustrates Kelvin's circulation theorem.

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• 3.I.8C

  Fluid Dynamics

  Part IB, 2003

  Show that the velocity field

  $$\mathbf{u}=\mathbf{U}+\frac{\boldsymbol{\Gamma} \times \mathbf{r}}{2 \pi r^{2}},$$

  where $\mathbf{U}=(U, 0,0), \mathbf{\Gamma}=(0,0, \Gamma)$ and $\mathbf{r}=(x, y, 0)$ in Cartesian coordinates $(x, y, z)$, represents the combination of a uniform flow and the flow due to a line vortex. Define and evaluate the circulation of the vortex.

  Show that

  $$\oint_{C_{R}}(\mathbf{u} \cdot \mathbf{n}) \mathbf{u} d l \rightarrow \frac{1}{2} \boldsymbol{\Gamma} \times \mathbf{U} \quad \text { as } \quad R \rightarrow \infty$$

  where $C_{R}$ is a circle $x^{2}+y^{2}=R^{2}, z=$ const. Explain how this result is related to the lift force on a two-dimensional aerofoil or other obstacle.

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• 3.II.18C

  Fluid Dynamics

  Part IB, 2003

  State the form of Bernoulli's theorem appropriate for an unsteady irrotational motion of an inviscid incompressible fluid in the absence of gravity.

  Water of density $\rho$ is driven through a tube of length $L$ and internal radius $a$ by the pressure exerted by a spherical, water-filled balloon of radius $R(t)$ attached to one end of the tube. The balloon maintains the pressure of the water entering the tube at $2 \gamma / R$ in excess of atmospheric pressure, where $\gamma$ is a constant. It may be assumed that the water exits the tube at atmospheric pressure. Show that

  $$R^{3} \ddot{R}+2 R^{2} \dot{R}^{2}=-\frac{\gamma a^{2}}{2 \rho L} .$$

  Solve equation ( $\dagger$ ), by multiplying through by $2 R \dot{R}$ or otherwise, to obtain

  $$t=R_{0}^{2}\left(\frac{2 \rho L}{\gamma a^{2}}\right)^{1 / 2}\left[\frac{\pi}{4}-\frac{\theta}{2}+\frac{1}{4} \sin 2 \theta\right]$$

  where $\theta=\sin ^{-1}\left(R / R_{0}\right)$ and $R_{0}$ is the initial radius of the balloon. Hence find the time when $R=0$.

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• 4.I.7C

  Fluid Dynamics

  Part IB, 2003

  Inviscid fluid issues vertically downwards at speed $u_{0}$ from a circular tube of radius a. The fluid falls onto a horizontal plate a distance $H$ below the end of the tube, where it spreads out axisymmetrically.

  Show that while the fluid is falling freely it has speed

  $$u=u_{0}\left[1+\frac{2 g}{u_{0}^{2}}(H-z)\right]^{1 / 2}$$

  and occupies a circular jet of radius

  $$R=a\left[1+\frac{2 g}{u_{0}^{2}}(H-z)\right]^{-1 / 4},$$

  where $z$ is the height above the plate and $g$ is the acceleration due to gravity.

  Show further that along the plate, at radial distances $r \gg a$ (i.e. far from the falling jet), where the fluid is flowing almost horizontally, it does so as a film of height $h(r)$, where

  $$\frac{a^{4}}{4 r^{2} h^{2}}=1+\frac{2 g}{u_{0}^{2}}(H-h)$$

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• 4.II.16C

  Fluid Dynamics

  Part IB, 2003

  Define the terms irrotational flow and incompressible flow. The two-dimensional flow of an incompressible fluid is given in terms of a streamfunction $\psi(x, y)$ as

  $$\mathbf{u}=(u, v)=\left(\frac{\partial \psi}{\partial y},-\frac{\partial \psi}{\partial x}\right)$$

  in Cartesian coordinates $(x, y)$. Show that the line integral

  $$\int_{\mathbf{x}_{1}}^{\mathbf{x}_{\mathbf{2}}} \mathbf{u} \cdot \mathbf{n} d l=\psi\left(\mathbf{x}_{\mathbf{2}}\right)-\psi\left(\mathbf{x}_{\mathbf{1}}\right)$$

  along any path joining the points $\mathbf{x}_{\mathbf{1}}$ and $\mathbf{x}_{\mathbf{2}}$, where $\mathbf{n}$ is the unit normal to the path. Describe how this result is related to the concept of mass conservation.

  Inviscid, incompressible fluid is contained in the semi-infinite channel $x>0$, $0<y<1$, which has rigid walls at $x=0$ and at $y=0,1$, apart from a small opening at the origin through which the fluid is withdrawn with volume flux $m$ per unit distance in the third dimension. Show that the streamfunction for irrotational flow in the channel can be chosen (up to an additive constant) to satisfy the equation

  $$\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}=0$$

  and boundary conditions

  if it is assumed that the flow at infinity is uniform. Solve the boundary-value problem above using separation of variables to obtain

  $$\psi=-m y+\frac{2 m}{\pi} \sum_{n=1}^{\infty} \frac{1}{n} \sin n \pi y e^{-n \pi x}$$

  $$\begin{aligned} & \psi=0 \quad \text { on } y=0, x>0, \\ & \psi=-m \quad \text { on } x=0,0<y<1, \\ & \psi=-m \quad \text { on } y=1, x>0, \\ & \psi \rightarrow-m y \quad \text { as } x \rightarrow \infty, \end{aligned}$$

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• $2 . \mathrm{I} . 4 \mathrm{E} \quad$

  Further Analysis

  Part IB, 2003

  Let $\tau_{1}$ be the collection of all subsets $A \subset \mathbb{N}$ such that $A=\emptyset$ or $\mathbb{N} \backslash A$ is finite. Let $\tau_{2}$ be the collection of all subsets of $\mathbb{N}$ of the form $I_{n}=\{n, n+1, n+2, \ldots\}$, together with the empty set. Prove that $\tau_{1}$ and $\tau_{2}$ are both topologies on $\mathbb{N}$.

  Show that a function $f$ from the topological space $\left(\mathbb{N}, \tau_{1}\right)$ to the topological space $\left(\mathbb{N}, \tau_{2}\right)$ is continuous if and only if one of the following alternatives holds:

  (i) $f(n) \rightarrow \infty$ as $n \rightarrow \infty$;

  (ii) there exists $N \in \mathbb{N}$ such that $f(n)=N$ for all but finitely many $n$ and $f(n) \leqslant N$ for all $n$.

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• 2.II.13E

  Further Analysis

  Part IB, 2003

  (a) Let $f:[1, \infty) \rightarrow \mathbb{C}$ be defined by $f(t)=t^{-1} e^{2 \pi i t}$ and let $X$ be the image of $f$. Prove that $X \cup\{0\}$ is compact and path-connected. [Hint: you may find it helpful to set $\left.s=t^{-1} .\right]$

  (b) Let $g:[1, \infty) \rightarrow \mathbb{C}$ be defined by $g(t)=\left(1+t^{-1}\right) e^{2 \pi i t}$, let $Y$ be the image of $g$ and let $\bar{D}$ be the closed unit $\operatorname{disc}\{z \in \mathbb{C}:|z| \leq 1\}$. Prove that $Y \cup \bar{D}$ is connected. Explain briefly why it is not path-connected.

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• $3 . \mathrm{I} . 3 \mathrm{E} \quad$

  Further Analysis

  Part IB, 2003

  (a) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an analytic function such that $|f(z)| \leqslant 1+|z|^{1 / 2}$ for every $z$. Prove that $f$ is constant.

  (b) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an analytic function such that $\operatorname{Re}(f(z)) \geqslant 0$ for every $z$. Prove that $f$ is constant.

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• 3.II.13E

  Further Analysis

  Part IB, 2003

  (a) State Taylor's Theorem.

  (b) Let $f(z)=\sum_{n=0}^{\infty} a_{n}\left(z-z_{0}\right)^{n}$ and $g(z)=\sum_{n=0}^{\infty} b_{n}\left(z-z_{0}\right)^{n}$ be defined whenever $\left|z-z_{0}\right|<r$. Suppose that $z_{k} \rightarrow z_{0}$ as $k \rightarrow \infty$, that no $z_{k}$ equals $z_{0}$ and that $f\left(z_{k}\right)=g\left(z_{k}\right)$ for every $k$. Prove that $a_{n}=b_{n}$ for every $n \geqslant 0$.

  (c) Let $D$ be a domain, let $z_{0} \in D$ and let $\left(z_{k}\right)$ be a sequence of points in $D$ that converges to $z_{0}$, but such that no $z_{k}$ equals $z_{0}$. Let $f: D \rightarrow \mathbb{C}$ and $g: D \rightarrow \mathbb{C}$ be analytic functions such that $f\left(z_{k}\right)=g\left(z_{k}\right)$ for every $k$. Prove that $f(z)=g(z)$ for every $z \in D$.

  (d) Let $D$ be the domain $\mathbb{C} \backslash\{0\}$. Give an example of an analytic function $f: D \rightarrow \mathbb{C}$ such that $f\left(n^{-1}\right)=0$ for every positive integer $n$ but $f$ is not identically 0 .

  (e) Show that any function with the property described in (d) must have an essential singularity at the origin.

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• 4.I.4E

  Further Analysis

  Part IB, 2003

  (a) State and prove Morera's Theorem.

  (b) Let $D$ be a domain and for each $n \in \mathbb{N}$ let $f_{n}: D \rightarrow \mathbb{C}$ be an analytic function. Suppose that $f: D \rightarrow \mathbb{C}$ is another function and that $f_{n} \rightarrow f$ uniformly on $D$. Prove that $f$ is analytic.

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• 4.II.13E

  Further Analysis

  Part IB, 2003

  (a) State the residue theorem and use it to deduce the principle of the argument, in a form that involves winding numbers.

  (b) Let $p(z)=z^{5}+z$. Find all $z$ such that $|z|=1$ and $\operatorname{Im}(p(z))=0$. Calculate $\operatorname{Re}(p(z))$ for each such $z$. [It will be helpful to set $z=e^{i \theta}$. You may use the addition formulae $\sin \alpha+\sin \beta=2 \sin \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)$ and $\cos \alpha+\cos \beta=2 \cos \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)$.]

  (c) Let $\gamma:[0,2 \pi] \rightarrow \mathbb{C}$ be the closed path $\theta \mapsto e^{i \theta}$. Use your answer to (b) to give a rough sketch of the path $p \circ \gamma$, paying particular attention to where it crosses the real axis.

  (d) Hence, or otherwise, determine for every real $t$ the number of $z$ (counted with multiplicity) such that $|z|<1$ and $p(z)=t$. (You need not give rigorous justifications for your calculations.)

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• 1.I.4F

  Geometry

  Part IB, 2003

  Describe the geodesics (that is, hyperbolic straight lines) in either the disc model or the half-plane model of the hyperbolic plane. Explain what is meant by the statements that two hyperbolic lines are parallel, and that they are ultraparallel.

  Show that two hyperbolic lines $l$ and $l^{\prime}$ have a unique common perpendicular if and only if they are ultraparallel.

  [You may assume standard results about the group of isometries of whichever model of the hyperbolic plane you use.]

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• 1.II.13F

  Geometry

  Part IB, 2003

  Write down the Riemannian metric in the half-plane model of the hyperbolic plane. Show that Möbius transformations mapping the upper half-plane to itself are isometries of this model.

  Calculate the hyperbolic distance from $i b$ to $i c$, where $b$ and $c$ are positive real numbers. Assuming that the hyperbolic circle with centre $i b$ and radius $r$ is a Euclidean circle, find its Euclidean centre and radius.

  Suppose that $a$ and $b$ are positive real numbers for which the points $i b$ and $a+i b$ of the upper half-plane are such that the hyperbolic distance between them coincides with the Euclidean distance. Obtain an expression for $b$ as a function of $a$. Hence show that, for any $b$ with $0<b<1$, there is a unique positive value of $a$ such that the hyperbolic distance between $i b$ and $a+i b$ coincides with the Euclidean distance.

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• 3.I.4F

  Geometry

  Part IB, 2003

  Show that any isometry of Euclidean 3 -space which fixes the origin can be written as a composite of at most three reflections in planes through the origin, and give (with justification) an example of an isometry for which three reflections are necessary.

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• 3.II.14F

  Geometry

  Part IB, 2003

  State and prove the Gauss-Bonnet formula for the area of a spherical triangle. Deduce a formula for the area of a spherical $n$-gon with angles $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$. For what range of values of $\alpha$ does there exist a (convex) regular spherical $n$-gon with angle $\alpha$ ?

  Let $\Delta$ be a spherical triangle with angles $\pi / p, \pi / q$ and $\pi / r$ where $p, q, r$ are integers, and let $G$ be the group of isometries of the sphere generated by reflections in the three sides of $\Delta$. List the possible values of $(p, q, r)$, and in each case calculate the order of the corresponding group $G$. If $(p, q, r)=(2,3,5)$, show how to construct a regular dodecahedron whose group of symmetries is $G$.

  [You may assume that the images of $\Delta$ under the elements of $G$ form a tessellation of the sphere.]

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• 1.I $. 5 \mathrm{E} \quad$

  Linear Mathematics

  Part IB, 2003

  Let $V$ be the subset of $\mathbb{R}^{5}$ consisting of all quintuples $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)$ such that

  $$a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=0$$

  and

  $$a_{1}+2 a_{2}+3 a_{3}+4 a_{4}+5 a_{5}=0$$

  Prove that $V$ is a subspace of $\mathbb{R}^{5}$. Solve the above equations for $a_{1}$ and $a_{2}$ in terms of $a_{3}, a_{4}$ and $a_{5}$. Hence, exhibit a basis for $V$, explaining carefully why the vectors you give form a basis.

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• 1.II.14E

  Linear Mathematics

  Part IB, 2003

  (a) Let $U, U^{\prime}$ be subspaces of a finite-dimensional vector space $V$. Prove that $\operatorname{dim}\left(U+U^{\prime}\right)=\operatorname{dim} U+\operatorname{dim} U^{\prime}-\operatorname{dim}\left(U \cap U^{\prime}\right) .$

  (b) Let $V$ and $W$ be finite-dimensional vector spaces and let $\alpha$ and $\beta$ be linear maps from $V$ to $W$. Prove that

  $$\operatorname{rank}(\alpha+\beta) \leqslant \operatorname{rank} \alpha+\operatorname{rank} \beta$$

  (c) Deduce from this result that

  $$\operatorname{rank}(\alpha+\beta) \geqslant|\operatorname{rank} \alpha-\operatorname{rank} \beta|$$

  (d) Let $V=W=\mathbb{R}^{n}$ and suppose that $1 \leqslant r \leqslant s \leqslant n$. Exhibit linear maps $\alpha, \beta: V \rightarrow W$ such that $\operatorname{rank} \alpha=r, \operatorname{rank} \beta=s$ and $\operatorname{rank}(\alpha+\beta)=s-r$. Suppose that $r+s \geqslant n$. Exhibit linear maps $\alpha, \beta: V \rightarrow W$ such that $\operatorname{rank} \alpha=r, \operatorname{rank} \beta=s$ and $\operatorname{rank}(\alpha+\beta)=n$.

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• 2.I.6E

  Linear Mathematics

  Part IB, 2003

  Let $a_{1}, a_{2}, \ldots, a_{n}$ be distinct real numbers. For each $i$ let $\mathbf{v}_{i}$ be the vector $\left(1, a_{i}, a_{i}^{2}, \ldots, a_{i}^{n-1}\right)$. Let $A$ be the $n \times n$ matrix with rows $\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n}$ and let $\mathbf{c}$ be a column vector of size $n$. Prove that $A \mathbf{c}=\mathbf{0}$ if and only if $\mathbf{c}=\mathbf{0}$. Deduce that the vectors $\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n} \operatorname{span} \mathbb{R}^{n}$.

  [You may use general facts about matrices if you state them clearly.]

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• 2.II.15E

  Linear Mathematics

  Part IB, 2003

  (a) Let $A=\left(a_{i j}\right)$ be an $m \times n$ matrix and for each $k \leqslant n$ let $A_{k}$ be the $m \times k$ matrix formed by the first $k$ columns of $A$. Suppose that $n>m$. Explain why the nullity of $A$ is non-zero. Prove that if $k$ is minimal such that $A_{k}$ has non-zero nullity, then the nullity of $A_{k}$ is 1 .

  (b) Suppose that no column of $A$ consists entirely of zeros. Deduce from (a) that there exist scalars $b_{1}, \ldots, b_{k}$ (where $k$ is defined as in (a)) such that $\sum_{j=1}^{k} a_{i j} b_{j}=0$ for every $i \leqslant m$, but whenever $\lambda_{1}, \ldots, \lambda_{k}$ are distinct real numbers there is some $i \leqslant m$ such that $\sum_{j=1}^{k} a_{i j} \lambda_{j} b_{j} \neq 0$.

  (c) Now let $\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{m}$ and $\mathbf{w}_{1}, \mathbf{w}_{2}, \ldots, \mathbf{w}_{m}$ be bases for the same real $m$ dimensional vector space. Let $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$ be distinct real numbers such that for every $j$ the vectors $\mathbf{v}_{1}+\lambda_{j} \mathbf{w}_{1}, \ldots, \mathbf{v}_{m}+\lambda_{j} \mathbf{w}_{m}$ are linearly dependent. For each $j$, let $a_{1 j}, \ldots, a_{m j}$ be scalars, not all zero, such that $\sum_{i=1}^{m} a_{i j}\left(\mathbf{v}_{i}+\lambda_{j} \mathbf{w}_{i}\right)=\mathbf{0}$. By applying the result of (b) to the matrix $\left(a_{i j}\right)$, deduce that $n \leqslant m$.

  (d) It follows that the vectors $\mathbf{v}_{1}+\lambda \mathbf{w}_{1}, \ldots, \mathbf{v}_{m}+\lambda \mathbf{w}_{m}$ are linearly dependent for at most $m$ values of $\lambda$. Explain briefly how this result can also be proved using determinants.

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• 3.I.7G

  Linear Mathematics

  Part IB, 2003

  Let $\alpha$ be an endomorphism of a finite-dimensional real vector space $U$ and let $\beta$ be another endomorphism of $U$ that commutes with $\alpha$. If $\lambda$ is an eigenvalue of $\alpha$, show that $\beta$ maps the kernel of $\alpha-\lambda \iota$ into itself, where $\iota$ is the identity map. Suppose now that $\alpha$ is diagonalizable with $n$ distinct real eigenvalues where $n=\operatorname{dim} U$. Prove that if there exists an endomorphism $\beta$ of $U$ such that $\alpha=\beta^{2}$, then $\lambda \geqslant 0$ for all eigenvalues $\lambda$ of $\alpha$.

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• 3.II.17G

  Linear Mathematics

  Part IB, 2003

  Define the determinant $\operatorname{det}(A)$ of an $n \times n$ complex matrix A. Let $A_{1}, \ldots, A_{n}$ be the columns of $A$, let $\sigma$ be a permutation of $\{1, \ldots, n\}$ and let $A^{\sigma}$ be the matrix whose columns are $A_{\sigma(1)}, \ldots, A_{\sigma(n)}$. Prove from your definition of determinant that $\operatorname{det}\left(A^{\sigma}\right)=\epsilon(\sigma) \operatorname{det}(A)$, where $\epsilon(\sigma)$ is the sign of the permutation $\sigma$. Prove also that $\operatorname{det}(A)=\operatorname{det}\left(A^{t}\right) .$

  Define the adjugate matrix $\operatorname{adj}(A)$ and prove from your $\operatorname{definitions}$ that $A \operatorname{adj}(A)=$ $\operatorname{adj}(A) A=\operatorname{det}(A) I$, where $I$ is the identity matrix. Hence or otherwise, prove that if $\operatorname{det}(A) \neq 0$, then $A$ is invertible.

  Let $C$ and $D$ be real $n \times n$ matrices such that the complex matrix $C+i D$ is invertible. By considering $\operatorname{det}(C+\lambda D)$ as a function of $\lambda$ or otherwise, prove that there exists a real number $\lambda$ such that $C+\lambda D$ is invertible. [You may assume that if a matrix $A$ is invertible, then $\operatorname{det}(A) \neq 0$.]

  Deduce that if two real matrices $A$ and $B$ are such that there exists an invertible complex matrix $P$ with $P^{-1} A P=B$, then there exists an invertible real matrix $Q$ such that $Q^{-1} A Q=B$.

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• 4.I.6G

  Linear Mathematics

  Part IB, 2003

  Let $\alpha$ be an endomorphism of a finite-dimensional real vector space $U$ such that $\alpha^{2}=\alpha$. Show that $U$ can be written as the direct sum of the kernel of $\alpha$ and the image of $\alpha$. Hence or otherwise, find the characteristic polynomial of $\alpha$ in terms of the dimension of $U$ and the rank of $\alpha$. Is $\alpha$ diagonalizable? Justify your answer.

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• 4.II.15G

  Linear Mathematics

  Part IB, 2003

  Let $\alpha \in L(U, V)$ be a linear map between finite-dimensional vector spaces. Let

  $$\begin{gathered} M^{l}(\alpha)=\{\beta \in L(V, U): \beta \alpha=0\} \quad \text { and } \\ M^{r}(\alpha)=\{\beta \in L(V, U): \alpha \beta=0\} . \end{gathered}$$

  (a) Prove that $M^{l}(\alpha)$ and $M^{r}(\alpha)$ are subspaces of $L(V, U)$ of dimensions

  $$\begin{gathered} \operatorname{dim} M^{l}(\alpha)=(\operatorname{dim} V-\operatorname{rank} \alpha) \operatorname{dim} U \quad \text { and } \\ \operatorname{dim} M^{r}(\alpha)=\operatorname{dim} \operatorname{ker}(\alpha) \operatorname{dim} V \end{gathered}$$

  [You may use the result that there exist bases in $U$ and $V$ so that $\alpha$ is represented by

  $$\left(\begin{array}{cc} I_{r} & 0 \\ 0 & 0 \end{array}\right)$$

  where $I_{r}$ is the $r \times r$ identity matrix and $r$ is the rank of $\left.\alpha .\right]$

  (b) Let $\Phi: L(U, V) \rightarrow L\left(V^{*}, U^{*}\right)$ be given by $\Phi(\alpha)=\alpha^{*}$, where $\alpha^{*}$ is the dual map induced by $\alpha$. Prove that $\Phi$ is an isomorphism. [You may assume that $\Phi$ is linear, and you may use the result that a finite-dimensional vector space and its dual have the same dimension.]

  (c) Prove that

  $$\Phi\left(M^{l}(\alpha)\right)=M^{r}\left(\alpha^{*}\right) \quad \text { and } \quad \Phi\left(M^{r}(\alpha)\right)=M^{l}\left(\alpha^{*}\right)$$

  [You may use the results that $(\beta \alpha)^{*}=\alpha^{*} \beta^{*}$ and that $\beta^{* *}$ can be identified with $\beta$ under the canonical isomorphism between a vector space and its double dual.]

  (d) Conclude that $\operatorname{rank}(\alpha)=\operatorname{rank}\left(\alpha^{*}\right)$.

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• 1.I.2D

  Methods

  Part IB, 2003

  Fermat's principle of optics states that the path of a light ray connecting two points will be such that the travel time $t$ is a minimum. If the speed of light varies continuously in a medium and is a function $c(y)$ of the distance from the boundary $y=0$, show that the path of a light ray is given by the solution to

  $$c(y) y^{\prime \prime}+c^{\prime}(y)\left(1+y^{\prime 2}\right)=0$$

  where $y^{\prime}=\frac{d y}{d x}$, etc. Show that the path of a light ray in a medium where the speed of light $c$ is a constant is a straight line. Also find the path from $(0,0)$ to $(1,0)$ if $c(y)=y$, and sketch it.

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• 1.II.11D

  Methods

  Part IB, 2003

  (a) Determine the Green's function $G(x, \xi)$ for the operator $\frac{d^{2}}{d x^{2}}+k^{2}$ on $[0, \pi]$ with Dirichlet boundary conditions by solving the boundary value problem

  $$\frac{d^{2} G}{d x^{2}}+k^{2} G=\delta(x-\xi), \quad G(0)=0, G(\pi)=0$$

  when $k$ is not an integer.

  (b) Use the method of Green's functions to solve the boundary value problem

  $$\frac{d^{2} y}{d x^{2}}+k^{2} y=f(x), \quad y(0)=a, y(\pi)=b$$

  when $k$ is not an integer.

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• 2.I.2C

  Methods

  Part IB, 2003

  Explain briefly why the second-rank tensor

  $$\int_{S} x_{i} x_{j} d S(\mathbf{x})$$

  is isotropic, where $S$ is the surface of the unit sphere centred on the origin.

  A second-rank tensor is defined by

  $$T_{i j}(\mathbf{y})=\int_{S}\left(y_{i}-x_{i}\right)\left(y_{j}-x_{j}\right) d S(\mathbf{x})$$

  where $S$ is the surface of the unit sphere centred on the origin. Calculate $T(\mathbf{y})$ in the form

  $$T_{i j}=\lambda \delta_{i j}+\mu y_{i} y_{j}$$

  where $\lambda$ and $\mu$ are to be determined.

  By considering the action of $T$ on $\mathbf{y}$ and on vectors perpendicular to $\mathbf{y}$, determine the eigenvalues and associated eigenvectors of $T$.

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• 2.II.11C

  Methods

  Part IB, 2003

  State the transformation law for an $n$ th-rank tensor $T_{i j \cdots k}$.

  Show that the fourth-rank tensor

  $$c_{i j k l}=\alpha \delta_{i j} \delta_{k l}+\beta \delta_{i k} \delta_{j l}+\gamma \delta_{i l} \delta_{j k}$$

  is isotropic for arbitrary scalars $\alpha, \beta$ and $\gamma$.

  The stress $\sigma_{i j}$ and strain $e_{i j}$ in a linear elastic medium are related by

  $$\sigma_{i j}=c_{i j k l} e_{k l} .$$

  Given that $e_{i j}$ is symmetric and that the medium is isotropic, show that the stress-strain relationship can be written in the form

  $$\sigma_{i j}=\lambda e_{k k} \delta_{i j}+2 \mu e_{i j}$$

  Show that $e_{i j}$ can be written in the form $e_{i j}=p \delta_{i j}+d_{i j}$, where $d_{i j}$ is a traceless tensor and $p$ is a scalar to be determined. Show also that necessary and sufficient conditions for the stored elastic energy density $E=\frac{1}{2} \sigma_{i j} e_{i j}$ to be non-negative for any deformation of the solid are that

  $$\mu \geq 0 \quad \text { and } \quad \lambda \geq-\frac{2}{3} \mu .$$

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• 3.I.2D

  Methods

  Part IB, 2003

  Consider the path between two arbitrary points on a cone of interior angle $2 \alpha$. Show that the arc-length of the path $r(\theta)$ is given by

  $$\int\left(r^{2}+r^{\prime 2} \operatorname{cosec}^{2} \alpha\right)^{1 / 2} d \theta$$

  where $r^{\prime}=\frac{d r}{d \theta}$. By minimizing the total arc-length between the points, determine the equation for the shortest path connecting them.

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• 3.II.12D

  Methods

  Part IB, 2003

  The transverse displacement $y(x, t)$ of a stretched string clamped at its ends $x=0, l$ satisfies the equation

  $$\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}-2 k \frac{\partial y}{\partial t}, \quad y(x, 0)=0, \frac{\partial y}{\partial t}(x, 0)=\delta(x-a)$$

  where $c>0$ is the wave velocity, and $k>0$ is the damping coefficient. The initial conditions correspond to a sharp blow at $x=a$ at time $t=0$.

  (a) Show that the subsequent motion of the string is given by

  $$y(x, t)=\frac{1}{\sqrt{\alpha_{n}^{2}-k^{2}}} \sum_{n} 2 e^{-k t} \sin \frac{\alpha_{n} a}{c} \sin \frac{\alpha_{n} x}{c} \sin /\left(\sqrt{\alpha_{n}^{2}-k^{2}} t\right)$$

  where $\alpha_{n}=\pi c n / l$.

  (b) Describe what happens in the limits of small and large damping. What critical parameter separates the two cases?

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• 4.I.2D

  Methods

  Part IB, 2003

  Consider the wave equation in a spherically symmetric coordinate system

  $$\frac{\partial^{2} u(r, t)}{\partial t^{2}}=c^{2} \Delta u(r, t)$$

  where $\Delta u=\frac{1}{r} \frac{\partial^{2}}{\partial r^{2}}(r u)$ is the spherically symmetric Laplacian operator.

  (a) Show that the general solution to the equation above is

  $$u(r, t)=\frac{1}{r}[f(r+c t)+g(r-c t)]$$

  where $f(x), g(x)$ are arbitrary functions.

  (b) Using separation of variables, determine the wave field $u(r, t)$ in response to a pulsating source at the origin $u(0, t)=A \sin \omega t$.

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• 4.II.11D

  Methods

  Part IB, 2003

  The velocity potential $\phi(r, \theta)$ for inviscid flow in two dimensions satisfies the Laplace equation

  $$\Delta \phi=\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}\right] \phi(r, \theta)=0$$

  (a) Using separation of variables, derive the general solution to the equation above that is single-valued and finite in each of the domains (i) $0 \leqslant r \leqslant a$; (ii) $a \leqslant r<\infty$.

  (b) Assuming $\phi$ is single-valued, solve the Laplace equation subject to the boundary conditions $\frac{\partial \phi}{\partial r}=0$ at $r=a$, and $\frac{\partial \phi}{\partial r} \rightarrow U \cos \theta$ as $r \rightarrow \infty$. Sketch the lines of constant potential.

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• 2.I.5B

  Numerical Analysis

  Part IB, 2003

  Let

  $$A=\left(\begin{array}{cccc} 1 & a & a^{2} & a^{3} \\ a^{3} & 1 & a & a^{2} \\ a^{2} & a^{3} & 1 & a \\ a & a^{2} & a^{3} & 1 \end{array}\right), \quad b=\left(\begin{array}{c} \gamma \\ 0 \\ 0 \\ \gamma a \end{array}\right), \quad \gamma=1-a^{4} \neq 0$$

  Find the LU factorization of the matrix $A$ and use it to solve the system $A x=b$.

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• 2.II.14B

  Numerical Analysis

  Part IB, 2003

  Let

  $$f^{\prime \prime}(0) \approx a_{0} f(-1)+a_{1} f(0)+a_{2} f(1)=\mu(f)$$

  be an approximation of the second derivative which is exact for $f \in \mathcal{P}_{2}$, the set of polynomials of degree $\leq 2$, and let

  $$e(f)=f^{\prime \prime}(0)-\mu(f)$$

  be its error.

  (a) Determine the coefficients $a_{0}, a_{1}, a_{2}$.

  (b) Using the Peano kernel theorem prove that, for $f \in C^{3}[-1,1]$, the set of threetimes continuously differentiable functions, the error satisfies the inequality

  $$|e(f)| \leq \frac{1}{3} \max _{x \in[-1,1]}\left|f^{\prime \prime \prime}(x)\right| .$$

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• 3.I.6B

  Numerical Analysis

  Part IB, 2003

  Given $(n+1)$ distinct points $x_{0}, x_{1}, \ldots, x_{n}$, let

  $$\ell_{i}(x)=\prod_{\substack{k=0 \\ k \neq i}}^{n} \frac{x-x_{k}}{x_{i}-x_{k}}$$

  be the fundamental Lagrange polynomials of degree $n$, let

  $$\omega(x)=\prod_{i=0}^{n}\left(x-x_{i}\right)$$

  and let $p$ be any polynomial of degree $\leq n$.

  (a) Prove that $\sum_{i=0}^{n} p\left(x_{i}\right) \ell_{i}(x) \equiv p(x)$.

  (b) Hence or otherwise derive the formula

  $$\frac{p(x)}{\omega(x)}=\sum_{i=0}^{n} \frac{A_{i}}{x-x_{i}}, \quad A_{i}=\frac{p\left(x_{i}\right)}{\omega^{\prime}\left(x_{i}\right)}$$

  which is the decomposition of $p(x) / \omega(x)$ into partial fractions.

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• 3.II.16B

  Numerical Analysis

  Part IB, 2003

  The functions $H_{0}, H_{1}, \ldots$ are generated by the Rodrigues formula:

  $$H_{n}(x)=(-1)^{n} e^{x^{2}} \frac{d^{n}}{d x^{n}} e^{-x^{2}}$$

  (a) Show that $H_{n}$ is a polynomial of degree $n$, and that the $H_{n}$ are orthogonal with respect to the scalar product

  $$(f, g)=\int_{-\infty}^{\infty} f(x) g(x) e^{-x^{2}} d x$$

  (b) By induction or otherwise, prove that the $H_{n}$ satisfy the three-term recurrence relation

  $$H_{n+1}(x)=2 x H_{n}(x)-2 n H_{n-1}(x) .$$

  [Hint: you may need to prove the equality $H_{n}^{\prime}(x)=2 n H_{n-1}(x)$ as well.]

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• $3 . \mathrm{I} . 5 \mathrm{H} \quad$

  Optimization

  Part IB, 2003

  Two players A and B play a zero-sum game with the pay-off matrix

  \begin{tabular}{r|rrr} & $B_{1}$ & $B_{2}$ & $B_{3}$ \ \hline$A_{1}$ & 4 & $-2$ & $-5$ \ $A_{2}$ & $-2$ & 4 & 3 \ $A_{3}$ & $-3$ & 6 & 2 \ $A_{4}$ & 3 & $-8$ & $-6$ \end{tabular}

  Here, the $(i, j)$ entry of the matrix indicates the pay-off to player A if he chooses move $A_{i}$ and player $\mathrm{B}$ chooses move $B_{j}$. Show that the game can be reduced to a zero-sum game with $2 \times 2$ pay-off matrix.

  Determine the value of the game and the optimal strategy for player A.

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• 3.II.15H

  Optimization

  Part IB, 2003

  Explain what is meant by a transportation problem where the total demand equals the total supply. Write the Lagrangian and describe an algorithm for solving such a problem. Starting from the north-west initial assignment, solve the problem with three sources and three destinations described by the table

  \begin{tabular}{|rrr|r|} \hline 5 & 9 & 1 & 36 \ 3 & 10 & 6 & 84 \ 7 & 2 & 5 & 40 \ \hline 14 & 68 & 78 & \ \hline \end{tabular}

  where the figures in the $3 \times 3$ box denote the transportation costs (per unit), the right-hand column denotes supplies, and the bottom row demands.

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• 4.I.5H

  Optimization

  Part IB, 2003

  State and prove the Lagrangian sufficiency theorem for a general optimization problem with constraints.

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• 4.II.14H

  Optimization

  Part IB, 2003

  Use the two-phase simplex method to solve the problem

  $$\begin{array}{llllll} \operatorname{minimize} & 5 x_{1}-12 x_{2}+13 x_{3} & & \\ \text { subject to } & 4 x_{1}+5 x_{2} & & \leq & 9 \\ & 6 x_{1}+4 x_{2}+ & x_{3} & \geq & 12 \\ & 3 x_{1}+2 x_{2}-x_{3} & \leq & 3 \\ & x_{i} \geq 0, & i=1,2,3 \end{array}$$

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• 1.I.8G

  Quadratic Mathematics

  Part IB, 2003

  Let $U$ and $V$ be finite-dimensional vector spaces. Suppose that $b$ and $c$ are bilinear forms on $U \times V$ and that $b$ is non-degenerate. Show that there exist linear endomorphisms $S$ of $U$ and $T$ of $V$ such that $c(x, y)=b(S(x), y)=b(x, T(y))$ for all $(x, y) \in U \times V$.

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• 1.II.17G

  Quadratic Mathematics

  Part IB, 2003

  (a) Suppose $p$ is an odd prime and $a$ an integer coprime to $p$. Define the Legendre symbol $\left(\frac{a}{p}\right)$ and state Euler's criterion.

  (b) Compute $\left(\frac{-1}{p}\right)$ and prove that

  $$\left(\frac{a b}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)$$

  whenever $a$ and $b$ are coprime to $p$.

  (c) Let $n$ be any integer such that $1 \leqslant n \leqslant p-2$. Let $m$ be the unique integer such that $1 \leqslant m \leqslant p-2$ and $m n \equiv 1(\bmod p)$. Prove that

  $$\left(\frac{n(n+1)}{p}\right)=\left(\frac{1+m}{p}\right)$$

  (d) Find

  $$\sum_{n=1}^{p-2}\left(\frac{n(n+1)}{p}\right)$$

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• 2.I.8G

  Quadratic Mathematics

  Part IB, 2003

  Let $U$ be a finite-dimensional real vector space and $b$ a positive definite symmetric bilinear form on $U \times U$. Let $\psi: U \rightarrow U$ be a linear map such that $b(\psi(x), y)+b(x, \psi(y))=0$ for all $x$ and $y$ in $U$. Prove that if $\psi$ is invertible, then the dimension of $U$ must be even. By considering the restriction of $\psi$ to its image or otherwise, prove that the rank of $\psi$ is always even.

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• 2.II.17G

  Quadratic Mathematics

  Part IB, 2003

  Let $S$ be the set of all $2 \times 2$ complex matrices $A$ which are hermitian, that is, $A^{*}=A$, where $A^{*}=\bar{A}^{t}$.

  (a) Show that $S$ is a real 4-dimensional vector space. Consider the real symmetric bilinear form $b$ on this space defined by

  $$b(A, B)=\frac{1}{2}(\operatorname{tr}(A B)-\operatorname{tr}(A) \operatorname{tr}(B)) .$$

  Prove that $b(A, A)=-\operatorname{det} A$ and $b(A, I)=-\frac{1}{2} \operatorname{tr}(A)$, where $I$ denotes the identity matrix.

  (b) Consider the three matrices

  $$A_{1}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right), \quad A_{2}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) \quad \text { and } \quad A_{3}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right)$$

  Prove that the basis $I, A_{1}, A_{2}, A_{3}$ of $S$ diagonalizes $b$. Hence or otherwise find the rank and signature of $b$.

  (c) Let $Q$ be the set of all $2 \times 2$ complex matrices $C$ which satisfy $C+C^{*}=\operatorname{tr}(C) I$. Show that $Q$ is a real 4-dimensional vector space. Given $C \in Q$, put

  $$\Phi(C)=\frac{1-i}{2} \operatorname{tr}(C) I+i C .$$

  Show that $\Phi$ takes values in $S$ and is a linear isomorphism between $Q$ and $S$.

  (d) Define a real symmetric bilinear form on $Q$ by setting $c(C, D)=-\frac{1}{2} \operatorname{tr}(C D)$, $C, D \in Q$. Show that $b(\Phi(C), \Phi(D))=c(C, D)$ for all $C, D \in Q$. Find the rank and signature of the symmetric bilinear form $c$ defined on $Q$.

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• 3.I.9G

  Quadratic Mathematics

  Part IB, 2003

  Let $f(x, y)=a x^{2}+b x y+c y^{2}$ be a binary quadratic form with integer coefficients. Explain what is meant by the discriminant $d$ of $f$. State a necessary and sufficient condition for some form of discriminant $d$ to represent an odd prime number $p$. Using this result or otherwise, find the primes $p$ which can be represented by the form $x^{2}+3 y^{2}$.

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• 3.II.19G

  Quadratic Mathematics

  Part IB, 2003

  Let $U$ be a finite-dimensional real vector space endowed with a positive definite inner product. A linear map $\tau: U \rightarrow U$ is said to be an orthogonal projection if $\tau$ is self-adjoint and $\tau^{2}=\tau$.

  (a) Prove that for every orthogonal projection $\tau$ there is an orthogonal decomposition

  $$U=\operatorname{ker}(\tau) \oplus \operatorname{im}(\tau)$$

  (b) Let $\phi: U \rightarrow U$ be a linear map. Show that if $\phi^{2}=\phi$ and $\phi \phi^{*}=\phi^{*} \phi$, where $\phi^{*}$ is the adjoint of $\phi$, then $\phi$ is an orthogonal projection. [You may find it useful to prove first that if $\phi \phi^{*}=\phi^{*} \phi$, then $\phi$ and $\phi^{*}$ have the same kernel.]

  (c) Show that given a subspace $W$ of $U$ there exists a unique orthogonal projection $\tau$ such that $\operatorname{im}(\tau)=W$. If $W_{1}$ and $W_{2}$ are two subspaces with corresponding orthogonal projections $\tau_{1}$ and $\tau_{2}$, show that $\tau_{2} \circ \tau_{1}=0$ if and only if $W_{1}$ is orthogonal to $W_{2}$.

  (d) Let $\phi: U \rightarrow U$ be a linear map satisfying $\phi^{2}=\phi$. Prove that one can define a positive definite inner product on $U$ such that $\phi$ becomes an orthogonal projection.

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• 1.I $. 9 \mathrm{~A} \quad$

  Quantum Mechanics

  Part IB, 2003

  A particle of mass $m$ is confined inside a one-dimensional box of length $a$. Determine the possible energy eigenvalues.

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• 1.II.18A

  Quantum Mechanics

  Part IB, 2003

  What is the significance of the expectation value

  $$\langle Q\rangle=\int \psi^{*}(x) Q \psi(x) d x$$

  of an observable $Q$ in the normalized state $\psi(x)$ ? Let $Q$ and $P$ be two observables. By considering the norm of $(Q+i \lambda P) \psi$ for real values of $\lambda$, show that

  $$\left\langle Q^{2}\right\rangle\left\langle P^{2}\right\rangle \geqslant \frac{1}{4}|\langle[Q, P]\rangle|^{2}$$

  The uncertainty $\Delta Q$ of $Q$ in the state $\psi(x)$ is defined as

  $$(\Delta Q)^{2}=\left\langle(Q-\langle Q\rangle)^{2}\right\rangle .$$

  Deduce the generalized uncertainty relation,

  $$\Delta Q \Delta P \geqslant \frac{1}{2}|\langle[Q, P]\rangle| .$$

  A particle of mass $m$ moves in one dimension under the influence of the potential $\frac{1}{2} m \omega^{2} x^{2}$. By considering the commutator $[x, p]$, show that the expectation value of the Hamiltonian satisfies

  $$\langle H\rangle \geqslant \frac{1}{2} \hbar \omega .$$

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• 2.I.9A

  Quantum Mechanics

  Part IB, 2003

  What is meant by the statement than an operator is hermitian?

  A particle of mass $m$ moves in the real potential $V(x)$ in one dimension. Show that the Hamiltonian of the system is hermitian.

  Show that

  $$\begin{aligned} \frac{d}{d t}\langle x\rangle &=\frac{1}{m}\langle p\rangle \\ \frac{d}{d t}\langle p\rangle &=\left\langle-V^{\prime}(x)\right\rangle \end{aligned}$$

  where $p$ is the momentum operator and $\langle A\rangle$ denotes the expectation value of the operator $A$.

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• 2.II.18A

  Quantum Mechanics

  Part IB, 2003

  A particle of mass $m$ and energy $E$ moving in one dimension is incident from the left on a potential barrier $V(x)$ given by

  $$V(x)= \begin{cases}V_{0} & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise }\end{cases}$$

  with $V_{0}>0$.

  In the limit $V_{0} \rightarrow \infty, a \rightarrow 0$ with $V_{0} a=U$ held fixed, show that the transmission probability is

  $$T=\left(1+\frac{m U^{2}}{2 E \hbar^{2}}\right)^{-1}$$

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• 3.II.20A

  Quantum Mechanics

  Part IB, 2003

  The radial wavefunction for the hydrogen atom satisfies the equation

  $$\frac{-\hbar^{2}}{2 m} \frac{1}{r^{2}} \frac{d}{d r}\left(r^{2} \frac{d}{d r} R(r)\right)+\frac{\hbar^{2}}{2 m r^{2}} \ell(\ell+1) R(r)-\frac{e^{2}}{4 \pi \epsilon_{0} r} R(r)=E R(r) .$$

  Explain the origin of each term in this equation.

  The wavefunctions for the ground state and first radially excited state, both with $\ell=0$, can be written as

  $$\begin{aligned} &R_{1}(r)=N_{1} \exp (-\alpha r) \\ &R_{2}(r)=N_{2}(r+b) \exp (-\beta r) \end{aligned}$$

  respectively, where $N_{1}$ and $N_{2}$ are normalization constants. Determine $\alpha, \beta, b$ and the corresponding energy eigenvalues $E_{1}$ and $E_{2}$.

  A hydrogen atom is in the first radially excited state. It makes the transition to the ground state, emitting a photon. What is the frequency of the emitted photon?

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• 3.I.10A

  Special Relativity

  Part IB, 2003

  What are the momentum and energy of a photon of wavelength $\lambda$ ?

  A photon of wavelength $\lambda$ is incident on an electron. After the collision, the photon has wavelength $\lambda^{\prime}$. Show that

  $$\lambda^{\prime}-\lambda=\frac{h}{m c}(1-\cos \theta),$$

  where $\theta$ is the scattering angle and $m$ is the electron rest mass.

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• 4.I.9A

  Special Relativity

  Part IB, 2003

  Prove that the two-dimensional Lorentz transformation can be written in the form

  $$\begin{aligned} x^{\prime} &=x \cosh \phi-c t \sinh \phi \\ c t^{\prime} &=-x \sinh \phi+c t \cosh \phi \end{aligned}$$

  where $\tanh \phi=v / c$. Hence, show that

  $$\begin{aligned} &x^{\prime}+c t^{\prime}=e^{-\phi}(x+c t) \\ &x^{\prime}-c t^{\prime}=e^{\phi}(x-c t) \end{aligned}$$

  Given that frame $S^{\prime}$ has speed $v$ with respect to $S$ and $S^{\prime \prime}$ has speed $v^{\prime}$ with respect to $S^{\prime}$, use this formalism to find the speed $v^{\prime \prime}$ of $S^{\prime \prime}$ with respect to $S$.

  [Hint: rotation through a hyperbolic angle $\phi$, followed by rotation through $\phi^{\prime}$, is equivalent to rotation through $\phi+\phi^{\prime}$.]

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• 4.II.18A

  Special Relativity

  Part IB, 2003

  A pion of rest mass $M$ decays at rest into a muon of rest mass $m<M$ and a neutrino of zero rest mass. What is the speed $u$ of the muon?

  In the pion rest frame $S$, the muon moves in the $y$-direction. A moving observer, in a frame $S^{\prime}$ with axes parallel to those in the pion rest frame, wishes to take measurements of the decay along the $x$-axis, and notes that the pion has speed $v$ with respect to the $x$-axis. Write down the four-dimensional Lorentz transformation relating $S^{\prime}$ to $S$ and determine the momentum of the muon in $S^{\prime}$. Hence show that in $S^{\prime}$ the direction of motion of the muon makes an angle $\theta$ with respect to the $y$-axis, where

  $$\tan \theta=\frac{M^{2}+m^{2}}{M^{2}-m^{2}} \frac{v}{\left(c^{2}-v^{2}\right)^{1 / 2}} .$$

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• $1 . \mathrm{I} . 3 \mathrm{H} \quad$

  Statistics

  Part IB, 2003

  Derive the least squares estimators $\hat{\alpha}$ and $\hat{\beta}$ for the coefficients of the simple linear regression model

  $$Y_{i}=\alpha+\beta\left(x_{i}-\bar{x}\right)+\varepsilon_{i}, \quad i=1, \ldots, n,$$

  where $x_{1}, \ldots, x_{n}$ are given constants, $\bar{x}=n^{-1} \sum_{i=1}^{n} x_{i}$, and $\varepsilon_{i}$ are independent with $\mathrm{E} \varepsilon_{i}=0, \operatorname{Var} \varepsilon_{i}=\sigma^{2}, i=1, \ldots, n$.

  A manufacturer of optical equipment has the following data on the unit cost (in pounds) of certain custom-made lenses and the number of units made in each order:

  \begin{tabular}{l|ccccc} No. of units, $x_{i}$ & 1 & 3 & 5 & 10 & 12 \ \hline Cost per unit, $y_{i}$ & 58 & 55 & 40 & 37 & 22 \end{tabular}

  Assuming that the conditions underlying simple linear regression analysis are met, estimate the regression coefficients and use the estimated regression equation to predict the unit cost in an order for 8 of these lenses.

  [Hint: for the data above, $S_{x y}=\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right) y_{i}=-257.4$.]

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• 1.II.12H

  Statistics

  Part IB, 2003

  Suppose that six observations $X_{1}, \ldots, X_{6}$ are selected at random from a normal distribution for which both the mean $\mu_{X}$ and the variance $\sigma_{X}^{2}$ are unknown, and it is found that $S_{X X}=\sum_{i=1}^{6}\left(x_{i}-\bar{x}\right)^{2}=30$, where $\bar{x}=\frac{1}{6} \sum_{i=1}^{6} x_{i}$. Suppose also that 21 observations $Y_{1}, \ldots, Y_{21}$ are selected at random from another normal distribution for which both the mean $\mu_{Y}$ and the variance $\sigma_{Y}^{2}$ are unknown, and it is found that $S_{Y Y}=40$. Derive carefully the likelihood ratio test of the hypothesis $H_{0}: \sigma_{X}^{2}=\sigma_{Y}^{2}$ against $H_{1}: \sigma_{X}^{2}>\sigma_{Y}^{2}$ and apply it to the data above at the $0.05$ level.

  

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• 2.I.3H

  Statistics

  Part IB, 2003

  Let $X_{1}, \ldots, X_{n}$ be a random sample from the $N\left(\theta, \sigma^{2}\right)$ distribution, and suppose that the prior distribution for $\theta$ is $N\left(\mu, \tau^{2}\right)$, where $\sigma^{2}, \mu, \tau^{2}$ are known. Determine the posterior distribution for $\theta$, given $X_{1}, \ldots, X_{n}$, and the best point estimate of $\theta$ under both quadratic and absolute error loss.

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• 2.II.12H

  Statistics

  Part IB, 2003

  An examination was given to 500 high-school students in each of two large cities, and their grades were recorded as low, medium, or high. The results are given in the table below.

  \begin{tabular}{l|ccc} & Low & Medium & High \ \hline City A & 103 & 145 & 252 \ City B & 140 & 136 & 224 \end{tabular}

  Derive carefully the test of homogeneity and test the hypothesis that the distributions of scores among students in the two cities are the same.

  

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• 4.I.3H

  Statistics

  Part IB, 2003

  The following table contains a distribution obtained in 320 tosses of 6 coins and the corresponding expected frequencies calculated with the formula for the binomial distribution for $p=0.5$ and $n=6$.

  \begin{tabular}{l|rrrrrrr} No. heads & 0 & 1 & 2 & 3 & 4 & 5 & 6 \ \hline Observed frequencies & 3 & 21 & 85 & 110 & 62 & 32 & 7 \ Expected frequencies & 5 & 30 & 75 & 100 & 75 & 30 & 5 \end{tabular}

  Conduct a goodness-of-fit test at the $0.05$ level for the null hypothesis that the coins are all fair.

  [Hint:

  $\left.\begin{array}{lcccc}\text { Distribution } & \chi_{5}^{2} & \chi_{6}^{2} & \chi_{7}^{2} \\ 95 \% \text { percentile } & 11.07 & 12.59 & 14.07 & \end{array}\right]$

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• 4.II.12H $\quad$

  Statistics

  Part IB, 2003

  State and prove the Rao-Blackwell theorem.

  Suppose that $X_{1}, \ldots, X_{n}$ are independent random variables uniformly distributed over $(\theta, 3 \theta)$. Find a two-dimensional sufficient statistic $T(X)$ for $\theta$. Show that an unbiased estimator of $\theta$ is $\hat{\theta}=X_{1} / 2$.

  Find an unbiased estimator of $\theta$ which is a function of $T(X)$ and whose mean square error is no more than that of $\hat{\theta}$.

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• A2.7

  Geometry of Surfaces

  Part II, 2003

  (i) What are geodesic polar coordinates at a point $P$ on a surface $M$ with a Riemannian metric $d s^{2}$ ?

  Assume that

  $$d s^{2}=d r^{2}+H(r, \theta)^{2} d \theta^{2}$$

  for geodesic polar coordinates $r, \theta$ and some function $H$. What can you say about $H$ and $d H / d r$ at $r=0$ ?

  (ii) Given that the Gaussian curvature $K$ may be computed by the formula $K=-H^{-1} \partial^{2} H / \partial r^{2}$, show that for small $R$ the area of the geodesic disc of radius $R$ centred at $P$ is

  $$\pi R^{2}-(\pi / 12) K R^{4}+a(R),$$

  where $a(R)$ is a function satisfying $\lim _{R \rightarrow 0} a(R) / R^{4}=0$.

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• A3.7

  Geometry of Surfaces

  Part II, 2003

  (i) Suppose that $C$ is a curve in the Euclidean $(\xi, \eta)$-plane and that $C$ is parameterized by its arc length $\sigma$. Suppose that $S$ in Euclidean $\mathbb{R}^{3}$ is the surface of revolution obtained by rotating $C$ about the $\xi$-axis. Take $\sigma, \theta$ as coordinates on $S$, where $\theta$ is the angle of rotation.

  Show that the Riemannian metric on $S$ induced from the Euclidean metric on $\mathbb{R}^{3}$ is

  $$d s^{2}=d \sigma^{2}+\eta(\sigma)^{2} d \theta^{2}$$

  (ii) For the surface $S$ described in Part (i), let $e_{\sigma}=\partial / \partial \sigma$ and $e_{\theta}=\partial / \partial \theta$. Show that, along any geodesic $\gamma$ on $S$, the quantity $g\left(\dot{\gamma}, e_{\theta}\right)$ is constant. Here $g$ is the metric tensor on $S$.

  [You may wish to compute $\left[X, e_{\theta}\right]=X e_{\theta}-e_{\theta} X$ for any vector field $X=A e_{\sigma}+B e_{\theta}$, where $A, B$ are functions of $\sigma, \theta$. Then use symmetry to compute $D_{\dot{\gamma}}\left(g\left(\dot{\gamma}, e_{\theta}\right)\right)$, which is the rate of change of $g\left(\dot{\gamma}, e_{\theta}\right)$ along $\gamma$.]

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• A4.7

  Geometry of Surfaces

  Part II, 2003

  Write an essay on the Theorema Egregium for surfaces in $\mathbb{R}^{3}$.

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• A1.8

  Graph Theory

  Part II, 2003

  (i) State Brooks' Theorem, and prove it in the case of a 3 -connected graph.

  (ii) Let $G$ be a bipartite graph, with vertex classes $X$ and $Y$, each of order $n$. If $G$ contains no cycle of length 4 show that

  $$e(G) \leqslant \frac{n}{2}(1+\sqrt{4 n-3})$$

  For which integers $n \leq 12$ are there examples where equality holds?

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• A $2 . 8$,

  Graph Theory

  Part II, 2003

  (i) State and prove a result of Euler relating the number of vertices, edges and faces of a plane graph. Use this result to exhibit a non-planar graph.

  (ii) State the vertex form of Menger's Theorem and explain how it follows from an appropriate version of the Max-flow-min-cut Theorem. Let $k \geqslant 2$. Show that every $k$-connected graph of order at least $2 k$ contains a cycle of length at least $2 k$.

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• A4.9

  Graph Theory

  Part II, 2003

  Write an essay on the vertex-colouring of graphs drawn on compact surfaces other than the sphere. You should include a proof of Heawood's bound, and an example of a surface for which this bound is not attained.

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• A1.9

  Number Theory

  Part II, 2003

  (i) Let $p$ be an odd prime and $k$ a strictly positive integer. Prove that the multiplicative group of relatively prime residue classes modulo $p^{k}$ is cyclic.

  [You may assume that the result is true for $k=1$.]

  (ii) Let $n=p_{1} p_{2} \ldots p_{r}$, where $r \geqslant 2$ and $p_{1}, p_{2}, \ldots, p_{r}$ are distinct odd primes. Let $B$ denote the set of all integers which are relatively prime to $n$. We recall that $n$ is said to be an Euler pseudo-prime to the base $b \in B$ if

  $$b^{(n-1) / 2} \equiv\left(\frac{b}{n}\right) \bmod n .$$

  If $n$ is an Euler pseudo-prime to the base $b_{1} \in B$, but is not an Euler pseudo-prime to the base $b_{2} \in B$, prove that $n$ is not an Euler pseudo-prime to the base $b_{1} b_{2}$. Let $p$ denote any of the primes $p_{1}, p_{2}, \ldots, p_{r}$. Prove that there exists a $b \in B$ such that

  $$\left(\frac{b}{p}\right)=-1 \quad \text { and } \quad b \equiv 1 \bmod n / p$$

  and deduce that $n$ is not an Euler pseudo-prime to this base $b$. Hence prove that $n$ is not an Euler pseudo-prime to the base $b$ for at least half of all the relatively prime residue classes $b \bmod n$.

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• A3.9

  Number Theory

  Part II, 2003

  (i) Let $x \geqslant 2$ be a real number and let $P(x)=\prod_{p \leqslant x}\left(1-\frac{1}{p}\right)^{-1}$, where the product is taken over all primes $p \leqslant x$. Prove that $P(x)>\log x$.

  (ii) Define the continued fraction of any positive irrational real number $x$. Illustrate your definition by computing the continued fraction of $1+\sqrt{3}$.

  Suppose that $a, b, c$ are positive integers with $b=a c$ and that $x$ has the periodic continued fraction $[b, a, b, a, \ldots]$. Prove that $x=\frac{1}{2}\left(b+\sqrt{b^{2}+4 c}\right)$.

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• A4.10

  Number Theory

  Part II, 2003

  Write an essay describing the factor base method for factorising a large odd positive integer $n$. Your essay should include a detailed explanation of how the continued fraction of $\sqrt{n}$ can be used to find a suitable factor base.

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• A2.9

  Coding and Cryptography

  Part II, 2003

  (i) Answer the following questions briefly but clearly.

  (a) How does coding theory apply when the error rate $p>1 / 2$ ?

  (b) Give an example of a code which is not a linear code.

  (c) Give an example of a linear code which is not a cyclic code.

  (d) Give an example of a general feedback register with output $k_{j}$, and initial fill $\left(k_{0}, k_{1}, \ldots, k_{N}\right)$, such that

  $$\left(k_{n}, k_{n+1}, \ldots, k_{n+N}\right) \neq\left(k_{0}, k_{1}, \ldots, k_{N}\right)$$

  for all $n \geqslant 1$.

  (e) Explain why the original Hamming code can not always correct two errors.

  (ii) Describe the Rabin-Williams scheme for coding a message $x$ as $x^{2}$ modulo a certain $N$. Show that, if $N$ is chosen appropriately, breaking this code is equivalent to factorising the product of two primes.

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• A2.10

  Algorithms and Networks

  Part II, 2003

  (i) Consider a network with node set $N$ and set of directed arcs $A$ equipped with functions $d^{+}: A \rightarrow \mathbb{Z}$ and $d^{-}: A \rightarrow \mathbb{Z}$ with $d^{-} \leqslant d^{+}$. Given $u: N \rightarrow \mathbb{R}$ we define the differential $\Delta u: A \rightarrow \mathbb{R}$ by $\Delta u(j)=u\left(i^{\prime}\right)-u(i)$ for $j=\left(i, i^{\prime}\right) \in A$. We say that $\Delta u$ is a feasible differential if

  $$d^{-}(j) \leqslant \Delta u(j) \leqslant d^{+}(j) \text { for all } j \in A$$

  Write down a necessary and sufficient condition on $d^{+}, d^{-}$for the existence of a feasible differential and prove its necessity.

  Assuming Minty's Lemma, describe an algorithm to construct a feasible differential and outline how this algorithm establishes the sufficiency of the condition you have given.

  (ii) Let $E \subseteq S \times T$, where $S, T$ are finite sets. A matching in $E$ is a subset $M \subseteq E$ such that, for all $s, s^{\prime} \in S$ and $t, t^{\prime} \in T$,

  $$\begin{array}{lll} (s, t),\left(s^{\prime}, t\right) \in M & \text { implies } & s=s^{\prime} \\ (s, t),\left(s, t^{\prime}\right) \in M & \text { implies } & t=t^{\prime} \end{array}$$

  A matching $M$ is maximal if for any other matching $M^{\prime}$ with $M \subseteq M^{\prime}$ we must have $M=M^{\prime}$. Formulate the problem of finding a maximal matching in $E$ in terms of an optimal distribution problem on a suitably defined network, and hence in terms of a standard linear optimization problem.

  [You may assume that the optimal distribution subject to integer constraints is integervalued.]

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• A3.10

  Algorithms and Networks

  Part II, 2003

  (i) Consider the problem

  $$\begin{aligned} \operatorname{minimize} & f(x) \\ \text { subject to } & h(x)=b, \quad x \in X, \end{aligned}$$

  where $f: \mathbb{R}^{n} \rightarrow \mathbb{R}, h: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}, X \subseteq \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$. State and prove the Lagrangian sufficiency theorem.

  In each of the following cases, where $n=2, m=1$ and $X=\{(x, y): x, y \geqslant 0\}$, determine whether the Lagrangian sufficiency theorem can be applied to solve the problem:

  (ii) Consider the problem in $\mathbb{R}^{n}$

  $$\begin{aligned} \operatorname{minimize} & \frac{1}{2} x^{T} Q x+c^{T} x \\ \text { subject to } & A x=b \end{aligned}$$

  where $Q$ is a positive-definite symmetric $n \times n$ matrix, $A$ is an $m \times n$ matrix, $c \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$. Explain how to reduce this problem to the solution of simultaneous linear equations.

  Consider now the problem

  $$\begin{aligned} \operatorname{minimize} & \frac{1}{2} x^{T} Q x+c^{T} x \\ \text { subject to } & A x \geqslant b \end{aligned}$$

  Describe the active set method for its solution.

  Consider the problem

  $$\begin{aligned} \operatorname{minimize} &(x-a)^{2}+(y-b)^{2}+x y \\ \text { subject to } & 0 \leqslant x \leqslant 1 \text { and } 0 \leqslant y \leqslant 1 \end{aligned}$$

  where $a, b \in \mathbb{R}$. Draw a diagram partitioning the $(a, b)$-plane into regions according to which constraints are active at the minimum.

  $$\begin{aligned} & \text { (a) } \quad f(x, y)=-x, \quad h(x, y)=x^{2}+y^{2}, \quad b=1 \text {; } \\ & \text { (b) } \quad f(x, y)=e^{-x y}, \quad h(x)=x, \quad b=0 \text {. } \end{aligned}$$

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• A1.10

  Coding and Cryptography

  Part II, 2003

  (i) We work over the field of two elements. Define what is meant by a linear code of length $n$. What is meant by a generator matrix for a linear code?

  Define what is meant by a parity check code of length $n$. Show that a code is linear if and only if it is a parity check code.

  Give the original Hamming code in terms of parity checks and then find a generator matrix for it.

  [You may use results from the theory of vector spaces provided that you quote them correctly.]

  (ii) Suppose that $1 / 4>\delta>0$ and let $\alpha(n, n \delta)$ be the largest information rate of any binary error correcting code of length $n$ which can correct $[n \delta]$ errors.

  Show that

  $$1-H(2 \delta) \leqslant \liminf _{n \rightarrow \infty} \alpha(n, n \delta) \leqslant 1-H(\delta)$$

  where

  $$H(\eta)=-\eta \log _{2} \eta-(1-\eta) \log _{2}(1-\eta)$$

  [You may assume any form of Stirling's theorem provided that you quote it correctly.]

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• A4.11

  Algorithms and Networks

  Part II, 2003

  Define the optimal distribution problem. State what it means for a circuit $P$ to be flow-augmenting, and what it means for $P$ to be unbalanced. State the optimality theorem for flows. Describe the simplex-on-a-graph algorithm, giving a brief justification of its stopping rules.

  Consider the problem of finding, in the network shown below, a minimum-cost flow from $s$ to $t$ of value 2 . Here the circled numbers are the upper arc capacities, the lower arc capacities all being zero, and the uncircled numbers are costs. Apply the simplex-on-agraph algorithm to solve this problem, taking as initial flow the superposition of a unit flow along the path $s,(s, a), a,(a, t), t$ and a unit flow along the path $s,(s, a), a,(a, b), b,(b, t), t$.

  

  Part II 2003

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• A1.13

  Computational Statistics and Statistical Modelling

  Part II, 2003

  (i) Suppose $Y_{i}, 1 \leqslant i \leqslant n$, are independent binomial observations, with $Y_{i} \sim B i\left(t_{i}, \pi_{i}\right)$, $1 \leqslant i \leqslant n$, where $t_{1}, \ldots, t_{n}$ are known, and we wish to fit the model

  $$\omega: \log \frac{\pi_{i}}{1-\pi_{i}}=\mu+\beta^{T} x_{i} \quad \text { for each } i$$

  where $x_{1}, \ldots, x_{n}$ are given covariates, each of dimension $p$. Let $\hat{\mu}, \hat{\beta}$ be the maximum likelihood estimators of $\mu, \beta$. Derive equations for $\hat{\mu}, \hat{\beta}$ and state without proof the form of the approximate distribution of $\hat{\beta}$.

  (ii) In 1975 , data were collected on the 3-year survival status of patients suffering from a type of cancer, yielding the following table

  \begin{tabular}{ccrr} & & \multicolumn{2}{c}{ survive? } \ age in years & malignant & yes & no \ under 50 & no & 77 & 10 \ under 50 & yes & 51 & 13 \ $50-69$ & no & 51 & 11 \ $50-69$ & yes & 38 & 20 \ $70+$ & no & 7 & 3 \ $70+$ & yes & 6 & 3 \end{tabular}

  Here the second column represents whether the initial tumour was not malignant or was malignant.

  Let $Y_{i j}$ be the number surviving, for age group $i$ and malignancy status $j$, for $i=1,2,3$ and $j=1,2$, and let $t_{i j}$ be the corresponding total number. Thus $Y_{11}=77$, $t_{11}=87$. Assume $Y_{i j} \sim B i\left(t_{i j}, \pi_{i j}\right), 1 \leqslant i \leqslant 3,1 \leqslant j \leqslant 2$. The results from fitting the model

  $$\log \left(\pi_{i j} /\left(1-\pi_{i j}\right)\right)=\mu+\alpha_{i}+\beta_{j}$$

  with $\alpha_{1}=0, \beta_{1}=0$ give $\hat{\beta}_{2}=-0.7328(\mathrm{se}=0.2985)$, and deviance $=0.4941$. What do you conclude?

  Why do we take $\alpha_{1}=0, \beta_{1}=0$ in the model?

  What "residuals" should you compute, and to which distribution would you refer them?

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• A2.12

  Computational Statistics and Statistical Modelling

  Part II, 2003

  (i) Suppose $Y_{1}, \ldots, Y_{n}$ are independent Poisson variables, and

  $$\mathbb{E}\left(Y_{i}\right)=\mu_{i}, \quad \log \mu_{i}=\alpha+\beta t_{i}, \quad \text { for } \quad i=1, \ldots, n,$$

  where $\alpha, \beta$ are two unknown parameters, and $t_{1}, \ldots, t_{n}$ are given covariates, each of dimension 1. Find equations for $\hat{\alpha}, \hat{\beta}$, the maximum likelihood estimators of $\alpha, \beta$, and show how an estimate of $\operatorname{var}(\hat{\beta})$ may be derived, quoting any standard theorems you may need.

  (ii) By 31 December 2001, the number of new vCJD patients, classified by reported calendar year of onset, were

  $$8,10,11,14,17,29,23$$

  for the years

  $$1994, \ldots, 2000 \text { respectively }$$

  Discuss carefully the (slightly edited) $R$ output for these data given below, quoting any standard theorems you may need.

  year

  year

  [1] 1994199519961997199819992000

  $>$ tot

  [1] $\begin{array}{lllllll}8 & 10 & 11 & 14 & 17 & 29 & 23\end{array}$

  first.glm - glm(tot year, family = poisson)

  $>\operatorname{summary}$ (first.glm)

  Call:

  glm(formula $=$ tot year, family $=$ poisson $)$

  Coefficients

  Estimate Std. Error z value $\operatorname{Pr}(>|z|)$

  (Intercept) $-407.8128599 .35366-4.1054 .05 \mathrm{e}-05$

  year $\quad 0.20556 \quad 0.04973 \quad 4.1333 .57 e-05$

  (Dispersion parameter for poisson family taken to be 1)

  Null deviance: $20.7753$ on 6 degrees of freedom

  Residual deviance: $2.7931$ on 5 degrees of freedom

  Number of Fisher Scoring iterations: 3

  Part II 2003

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• A4.14

  Computational Statistics and Statistical Modelling

  Part II, 2003

  The nave height $x$, and the nave length $y$ for 16 Gothic-style cathedrals and 9 Romanesque-style cathedrals, all in England, have been recorded, and the corresponding $R$ output (slightly edited) is given below.

  

  You may assume that $x, y$ are in suitable units, and that "style" has been set up as a factor with levels 1,2 corresponding to Gothic, Romanesque respectively.

  (a) Explain carefully, with suitable graph(s) if necessary, the results of this analysis.

  (b) Using the general model $Y=X \beta+\epsilon$ (in the conventional notation) explain carefully the theory needed for (a).

  [Standard theorems need not be proved.]

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• A1.14

  Quantum Physics

  Part II, 2003

  (i) An electron of mass $m$ and spin $\frac{1}{2}$ moves freely inside a cubical box of side $L$. Verify that the energy eigenstates of the system are $\phi_{l m n}(\mathbf{r}) \chi_{\pm}$where the spatial wavefunction is given by

  $$\phi_{l m n}(\mathbf{r})=\left(\frac{2}{L}\right)^{3 / 2} \sin \left(\frac{l \pi x}{L}\right) \sin \left(\frac{m \pi y}{L}\right) \sin \left(\frac{n \pi z}{L}\right)$$

  and

  $$\chi_{+}=\left(\begin{array}{l} 1 \\ 0 \end{array}\right), \quad \chi_{-}=\left(\begin{array}{l} 0 \\ 1 \end{array}\right)$$

  Give the corresponding energy eigenvalues.

  A second electron is inserted into the box. Explain how the Pauli principle determines the structure of the wavefunctions associated with the lowest energy level and the first excited energy level. What are the values of the energy in these two levels and what are the corresponding degeneracies?

  (ii) When the side of the box, $L$, is large, the number of eigenstates available to the electron with energy in the range $E \rightarrow E+d E$ is $\rho(E) d E$. Show that

  $$\rho(E)=\frac{L^{3}}{\pi^{2} \hbar^{3}} \sqrt{2 m^{3} E}$$

  A large number, $N$, of electrons are inserted into the box. Explain how the ground state is constructed and define the Fermi energy, $E_{F}$. Show that in the ground state

  $$N=\frac{2}{3} \frac{L^{3}}{\pi^{2} \hbar^{3}} \sqrt{2 m^{3}}\left(E_{F}\right)^{3 / 2}$$

  When a magnetic field $H$ in the $z$-direction is applied to the system, an electron with spin up acquires an additional energy $+\mu H$ and an electron with spin down an energy $-\mu H$, where $-\mu$ is the magnetic moment of the electron and $\mu>0$. Describe, for the case $E_{F}>\mu H$, the structure of the ground state of the system of $N$ electrons in the box and show that

  $$N=\frac{1}{3} \frac{L^{3}}{\pi^{2} \hbar^{3}} \sqrt{2 m^{3}}\left(\left(E_{F}+\mu H\right)^{3 / 2}+\left(E_{F}-\mu H\right)^{3 / 2}\right) \text {. }$$

  Calculate the induced magnetic moment, $M$, of the ground state of the system and show that for a weak magnetic field the magnetic moment is given by

  $$M \approx \frac{3}{2} N \frac{\mu^{2} H}{E_{F}}$$

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• A2.14

  Quantum Physics

  Part II, 2003

  (i) A system of $N$ distinguishable non-interacting particles has energy levels $E_{i}$ with degeneracy $g_{i}, 1 \leqslant i<\infty$, for each particle. Show that in thermal equilibrium the number of particles $N_{i}$ with energy $E_{i}$ is given by

  $$N_{i}=g_{i} e^{-\beta\left(E_{i}-\mu\right)}$$

  where $\beta$ and $\mu$ are parameters whose physical significance should be briefly explained.

  A gas comprises a set of atoms with non-degenerate energy levels $E_{i}, 1 \leqslant i<\infty$. Assume that the gas is dilute and the motion of the atoms can be neglected. For such a gas the atoms can be treated as distinguishable. Show that when the system is at temperature $T$, the number of atoms $N_{i}$ at level $i$ and the number $N_{j}$ at level $j$ satisfy

  $$\frac{N_{i}}{N_{j}}=e^{-\left(E_{i}-E_{j}\right) / k T}$$

  where $k$ is Boltzmann's constant.

  (ii) A system of bosons has a set of energy levels $W_{a}$ with degeneracy $f_{a}, 1 \leqslant a<\infty$, for each particle. In thermal equilibrium at temperature $T$ the number $n_{a}$ of particles in level $a$ is

  $$n_{a}=\frac{f_{a}}{e^{\left(W_{a}-\mu\right) / k T}-1} \text {. }$$

  What is the value of $\mu$ when the particles are photons?

  Given that the density of states $\rho(\omega)$ for photons of frequency $\omega$ in a cubical box of side $L$ is

  $$\rho(\omega)=L^{3} \frac{\omega^{2}}{\pi^{2} c^{3}}$$

  where $c$ is the speed of light, show that at temperature $T$ the density of photons in the frequency range $\omega \rightarrow \omega+d \omega$ is $n(\omega) d \omega$ where

  $$n(\omega)=\frac{\omega^{2}}{\pi^{2} c^{3}} \frac{1}{e^{\hbar \omega / k T}-1}$$

  Deduce the energy density, $\epsilon(\omega)$, for photons of frequency $\omega$.

  The cubical box is occupied by the gas of atoms described in Part (i) in the presence of photons at temperature $T$. Consider the two atomic levels $i$ and $j$ where $E_{i}>E_{j}$ and $E_{i}-E_{j}=\hbar \omega$. The rate of spontaneous photon emission for the transition $i \rightarrow j$ is $A_{i j}$. The rate of absorption is $B_{j i} \epsilon(\omega)$ and the rate of stimulated emission is $B_{i j} \epsilon(\omega)$. Show that the requirement that these processes maintain the atoms and photons in thermal equilibrium yields the relations

  $$B_{i j}=B_{j i}$$

  and

  $$A_{i j}=\left(\frac{\hbar \omega^{3}}{\pi^{2} c^{3}}\right) B_{i j}$$

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• A4.16

  Quantum Physics

  Part II, 2003

  Describe the energy band structure available to electrons moving in crystalline materials. How can it be used to explain the properties of crystalline materials that are conductors, insulators and semiconductors?

  Where does the Fermi energy lie in an intrinsic semiconductor? Describe the process of doping of semiconductors and explain the difference between $n$-type and $p$-type doping. What is the effect of the doping on the position of the Fermi energy in the two cases?

  Why is there a potential difference across a junction of $n$-type and $p$-type semiconductors?

  Derive the relation

  $$I=I_{0}\left(1-e^{-q V / k T}\right)$$

  between the current, $I$, and the voltage, $V$, across an $n p$ junction, where $I_{0}$ is the total minority current in the semiconductor and $q$ is the charge on the electron, $T$ is the temperature and $k$ is Boltzmann's constant. Your derivation should include an explanation of the terms majority current and minority current.

  Why can the $n p$ junction act as a rectifier?

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• A1.16

  Statistical Physics and Cosmology

  Part II, 2003

  (i) Explain briefly how the relative motion of galaxies in a homogeneous and isotropic universe is described in terms of the scale factor $a(t)$ (where $t$ is time). In particular, show that the relative velocity $\mathbf{v}(t)$ of two galaxies is given in terms of their relative displacement $\mathbf{r}(t)$ by the formula $\mathbf{v}(t)=H(t) \mathbf{r}(t)$, where $H(t)$ is a function that you should determine in terms of $a(t)$. Given that $a(0)=0$, obtain a formula for the distance $R(t)$ to the cosmological horizon at time $t$. Given further that $a(t)=\left(t / t_{0}\right)^{\alpha}$, for $0<\alpha<1$ and constant $t_{0}$, compute $R(t)$. Hence show that $R(t) / a(t) \rightarrow 0$ as $t \rightarrow 0$.

  (ii) A homogeneous and isotropic model universe has energy density $\rho(t) c^{2}$ and pressure $P(t)$, where $c$ is the speed of light. The evolution of its scale factor $a(t)$ is governed by the Friedmann equation

  $$\dot{a}^{2}=\frac{8 \pi G}{3} \rho a^{2}-k c^{2}$$

  where the overdot indicates differentiation with respect to $t$. Use the "Fluid" equation

  $$\dot{\rho}=-3\left(\rho+\frac{P}{c^{2}}\right)\left(\frac{\dot{a}}{a}\right)$$

  to obtain an equation for the acceleration $\ddot{a}(t)$. Assuming $\rho>0$ and $P \geqslant 0$, show that $\rho a^{3}$ cannot increase with time as long as $\dot{a}>0$, nor decrease if $\dot{a}<0$. Hence determine the late time behaviour of $a(t)$ for $k<0$. For $k>0$ show that an initially expanding universe must collapse to a "big crunch" at which $a \rightarrow 0$. How does $\dot{a}$ behave as $a \rightarrow 0$ ? Given that $P=0$, determine the form of $a(t)$ near the big crunch. Discuss the qualitative late time behaviour for $k=0$.

  Cosmological models are often assumed to have an equation of state of the form $P=\sigma \rho c^{2}$ for constant $\sigma$. What physical principle requires $\sigma \leqslant 1$ ? Matter with $P=\rho c^{2}$ $(\sigma=1)$ is called "stiff matter" by cosmologists. Given that $k=0$, determine $a(t)$ for a universe that contains only stiff matter. In our Universe, why would you expect stiff matter to be negligible now even if it were significant in the early Universe?

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• A3.14

  Statistical Physics and Cosmology

  Part II, 2003

  (i) The pressure $P(r)$ and mass density $\rho(r)$, at distance $r$ from the centre of a spherically-symmetric star, obey the pressure-support equation

  $$P^{\prime}=-\frac{G m \rho}{r^{2}}$$

  where $m^{\prime}=4 \pi r^{2} \rho(r)$, and the prime indicates differentiation with respect to $r$. Let $V$ be the total volume of the star, and $\langle P\rangle$ its average pressure. Use the pressure-support equation to derive the "virial theorem"

  $$\langle P\rangle V=-\frac{1}{3} E_{g r a v}$$

  where $E_{g r a v}$ is the total gravitational potential energy [Hint: multiply by $4 \pi r^{3}$ ]. If a star is assumed to be a self-gravitating ball of a non-relativistic ideal gas then it can be shown that

  $$\langle P\rangle V=\frac{2}{3} E_{k i n}$$

  where $E_{k i n}$ is the total kinetic energy. Use this result to show that the total energy $U=E_{\text {grav }}+E_{k i n}$ is negative. When nuclear reactions have converted the hydrogen in a star's core to helium the core contracts until the helium is converted to heavier elements, thereby increasing the total energy $U$ of the star. Explain briefly why this converts the star into a "Red Giant". (ii) Write down the first law of thermodynamics for the change in energy $E$ of a system at temperature $T$, pressure $P$ and chemical potential $\mu$ as a result of small changes in the entropy $S$, volume $V$ and particle number $N$. Use this to show that

  $$P=-\left(\frac{\partial E}{\partial V}\right)_{N, S}$$

  The microcanonical ensemble is the set of all accessible microstates of a system at fixed $E, V, N$. Define the canonical and grand-canonical ensembles. Why are the properties of a macroscopic system independent of the choice of thermodynamic ensemble?

  The Gibbs "grand potential" $\mathcal{G}(T, V, \mu)$ can be defined as

  $$\mathcal{G}=E-T S-\mu N$$

  Use the first law to find expressions for $S, P, N$ as partial derivatives of $\mathcal{G}$. A system with variable particle number $n$ has non-degenerate energy eigenstates labeled by $r^{(n)}$, for each $n$, with energy eigenvalues $E_{r}^{(n)}$. If the system is in equilibrium at temperature $T$ and chemical potential $\mu$ then the probability $p\left(r^{(n)}\right)$ that it will be found in a particular $n$-particle state $r^{(n)}$ is given by the Gibbs probability distribution

  $$p\left(r^{(n)}\right)=\mathcal{Z}^{-1} e^{\left(\mu n-E_{r}^{(n)}\right) / k T}$$

  where $k$ is Boltzmann's constant. Deduce an expression for the normalization factor $\mathcal{Z}$ as a function of $\mu$ and $\beta=1 / k T$, and hence find expressions for the partial derivatives

  $$\frac{\partial \log \mathcal{Z}}{\partial \mu}, \quad \frac{\partial \log \mathcal{Z}}{\partial \beta}$$

  in terms of $N, E, \mu, \beta$.

  Why does $\mathcal{Z}$ also depend on the volume $V$ ? Given that a change in $V$ at fixed $N, S$ leaves unchanged the Gibbs probability distribution, deduce that

  $$\left(\frac{\partial \log \mathcal{Z}}{\partial V}\right)_{\mu, \beta}=\beta P$$

  Use your results to show that

  $$\mathcal{G}=-k T \log \left(\mathcal{Z} / \mathcal{Z}_{0}\right)$$

  for some constant $\mathcal{Z}_{0}$.

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• A4.18

  Statistical Physics and Cosmology

  Part II, 2003

  Let $g(p)$ be the density of states of a particle in volume $V$ as a function of the magnitude $p$ of the particle's momentum. Explain why $g(p) \propto V p^{2} / h^{3}$, where $h$ is Planck's constant. Write down the Bose-Einstein and Fermi-Dirac distributions for the (average) number $\bar{n}(p)$ of particles of an ideal gas with momentum $p$. Hence write down integrals for the (average) total number $N$ of particles and the (average) total energy $E$ as functions of temperature $T$ and chemical potential $\mu$. Why do $N$ and $E$ also depend on the volume $V ?$

  Electromagnetic radiation in thermal equilibrium can be regarded as a gas of photons. Why are photons "ultra-relativistic" and how is photon momentum $p$ related to the frequency $\nu$ of the radiation? Why does a photon gas have zero chemical potential? Use your formula for $\bar{n}(p)$ to express the energy density $\varepsilon_{\gamma}$ of electromagnetic radiation in the form

  $$\varepsilon_{\gamma}=\int_{0}^{\infty} \epsilon(\nu) d \nu$$

  where $\epsilon(\nu)$ is a function of $\nu$ that you should determine up to a dimensionless multiplicative constant. Show that $\epsilon(\nu)$ is independent of $h$ when $k T \gg h \nu$, where $k$ is Boltzmann's constant. Let $\nu_{\text {peak }}$ be the value of $\nu$ at the maximum of the function $\epsilon(\nu)$; how does $\nu_{\text {peak }}$ depend on $T$ ?

  Let $n_{\gamma}$ be the photon number density at temperature $T$. Show that $n_{\gamma} \propto T^{q}$ for some power $q$, which you should determine. Why is $n_{\gamma}$ unchanged as the volume $V$ is increased quasi-statically? How does $T$ depend on $V$ under these circumstances? Applying your result to the Cosmic Microwave Background Radiation (CMBR), deduce how the temperature $T_{\gamma}$ of the CMBR depends on the scale factor $a$ of the Universe. At a time when $T_{\gamma} \sim 3000 K$, the Universe underwent a transition from an earlier time at which it was opaque to a later time at which it was transparent. Explain briefly the reason for this transition and its relevance to the CMBR.

  An ideal gas of fermions $f$ of mass $m$ is in equilibrium at temperature $T$ and chemical potential $\mu_{f}$ with a gas of its own anti-particles $\bar{f}$ and photons $(\gamma)$. Assuming that chemical equilibrium is maintained by the reaction

  $$f+\bar{f} \leftrightarrow \gamma$$

  determine the chemical potential $\mu_{\bar{f}}$ of the antiparticles. Let $n_{f}$ and $n_{\bar{f}}$ be the number densities of $f$ and $\bar{f}$, respectively. What will their values be for $k T \ll m c^{2}$ if $\mu_{f}=0$ ? Given that $\mu_{f}>0$, but $\mu_{f} \ll k T$, show that

  $$n_{f} \approx n_{0}(T)\left[1+\frac{\mu_{f}}{k T} F\left(m c^{2} / k T\right)\right]$$

  where $n_{0}(T)$ is the fermion number density at zero chemical potential and $F$ is a positive function of the dimensionless ratio $m c^{2} / k T$. What is $F$ when $k T \ll m c^{2}$ ?

  Given that $\mu_{f} \ll k T$, obtain an expression for the ratio $\left(n_{f}-n_{\bar{f}}\right) / n_{0}$ in terms of $\mu, T$ and the function $F$. Supposing that $f$ is either a proton or neutron, why should you expect the ratio $\left(n_{f}-n_{\bar{f}}\right) / n_{\gamma}$ to remain constant as the Universe expands?

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• A1.17

  Symmetries and Groups in Physics

  Part II, 2003

  (i) Define the character $\chi$ of a representation $D$ of a finite group $G$. Show that $<\chi \mid \chi>=1$ if and only if $D$ is irreducible, where

  $$<\chi \mid \chi>=\frac{1}{|G|} \sum_{g \in G} \chi(g) \chi\left(g^{-1}\right)$$

  If $|G|=8$ and $<\chi \mid \chi>=2$, what are the possible dimensions of the representation $D ?$

  (ii) State and prove Schur's first and second lemmas.

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• A3.15

  Symmetries and Groups in Physics

  Part II, 2003

  (i) Given that the character of an $S U(2)$ transformation in the $(2 l+1)$-dimensional irreducible representation $d_{l}$ is given by

  $$\chi_{l}(\theta)=\frac{\sin \left(l+\frac{1}{2}\right) \theta}{\sin \frac{\theta}{2}}$$

  show how the direct product representation $d_{l_{1}} \otimes d_{l_{2}}$ decomposes into irreducible $S U(2)$ representations.

  (ii) Find the decomposition of the direct product representation $3 \otimes \overline{3}$ of $S U(3)$ into irreducible $S U(3)$ representations.

  Mesons consist of one quark and one antiquark. The scalar Meson Octet consists of the following particles: $K^{\pm}\left(Y=\pm 1, I_{3}=\pm \frac{1}{2}\right), K^{0}\left(Y=1, I_{3}=-\frac{1}{2}\right), \bar{K}^{0}(Y=-1$, $\left.I_{3}=\frac{1}{2}\right), \pi^{\pm}\left(Y=0, I_{3}=\pm 1\right), \pi^{0}\left(Y=0, I_{3}=0\right)$ and $\eta\left(Y=0, I_{3}=0\right)^{2}$.

  Use the direct product representation $3 \otimes \overline{3}$ of $S U(3)$ to identify the quark-type of the particles in the scalar Meson Octet. Deduce the quark-type of the $S U(3)$ singlet state $\eta^{\prime}$ contained in $3 \otimes \overline{3}$.

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• A1.18

  Transport Processes

  Part II, 2003

  (i) A solute occupying a domain $V_{0}$ has concentration $C(\boldsymbol{x}, t)$ and is created at a rate $S(\boldsymbol{x}, t)$ per unit volume; $\boldsymbol{J}(\boldsymbol{x}, t)$ is the flux of solute per unit area; $\boldsymbol{x}, t$ are position and time. Derive the transport equation

  $$C_{t}+\nabla \cdot \boldsymbol{J}=S$$

  State Fick's Law of diffusion and hence write down the diffusion equation for $C(\boldsymbol{x}, t)$ for a case in which the solute flux occurs solely by diffusion, with diffusivity $D(\boldsymbol{x})$.

  In a finite domain $0 \leqslant x \leqslant L, D, S$ and the steady-state distribution of $C$ depend only on $x ; C$ is equal to $C_{0}$ at $x=0$ and $C_{1} \neq C_{0}$ at $x=L$. Find $C(x)$ in the following two cases: (a) $D=D_{0}, S=0$, (b) $D=D_{1} x^{1 / 2}, S=0$,

  where $D_{0}$ and $D_{1}$ are positive constants.

  Show that there is no steady solution satisfying the boundary conditions if $D=$ $D_{1} x, S=0 .$

  (ii) For the problem of Part (i), consider the case $D=D_{0}, S=k C$, where $D_{0}$ and $k$ are positive constants. Calculate the steady-state solution, $C=C_{s}(x)$, assuming that $\sqrt{k / D_{0}} \neq n \pi / L$ for any integer $n$.

  Now let

  $$C(x, 0)=C_{0} \frac{\sin \alpha(L-x)}{\sin \alpha L}$$

  where $\alpha=\sqrt{k / D_{0}}$. Find the equations, boundary and initial conditions satisfied by $C^{\prime}(x, t)=C(x, t)-C_{s}(x)$. Solve the problem using separation of variables and show that

  $$C^{\prime}(x, t)=\sum_{n=1}^{\infty} A_{n} \sin \frac{n \pi x}{L} \exp \left[\left(\alpha^{2}-\frac{n^{2} \pi^{2}}{L^{2}}\right) D_{0} t\right]$$

  for some constants $A_{n}$. Write down an integral expression for $A_{n}$, show that

  $$A_{1}=-\frac{2 \pi C_{1}}{\alpha^{2} L^{2}-\pi^{2}},$$

  and comment on the behaviour of the solution for large times in the two cases $\alpha L<\pi$ and $\alpha L>\pi$.

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• A3.16

  Transport Processes

  Part II, 2003

  (i) When a solid crystal grows into a supercooled infinite melt, latent heat must be removed from the interface by diffusion into the melt. Write down the equation and boundary conditions satisfied by the temperature $\theta(\boldsymbol{x}, t)$ in the melt, where $\boldsymbol{x}$ is position and $t$ time, in terms of the following material properties: solid density $\rho_{s}$, specific heat capacity $C_{p}$, coefficient of latent heat per unit mass $L$, thermal conductivity $k$, melting temperature $\theta_{m}$. You may assume that the densities of the melt and the solid are the same and that temperature in the melt far from the interface is $\theta_{m}-\Delta \theta$, where $\Delta \theta$ is a positive constant.

  A spherical crystal of radius $a(t)$ grows into such a melt with $a(0)=0$. Use dimensional analysis to show that $a(t)$ is proportional to $t^{1 / 2}$.

  (ii) Show that the above problem should have a similarity solution of the form

  $$\theta=\theta_{m}-\Delta \theta(1-F(\xi))$$

  where $\xi=r(\kappa t)^{-1 / 2}, r$ is the radial coordinate in spherical polars and $\kappa=k / \rho_{s} C_{p}$ is the thermal diffusivity. Recalling that, for spherically symmetric $\theta, \nabla^{2} \theta=\frac{1}{r^{2}}\left(r^{2} \theta_{r}\right)_{r}$, write down the equation and boundary conditions to be satisfied by $F(\xi)$. Hence show that the radius of the crystal is given by $a(t)=\lambda(\kappa t)^{1 / 2}$, where $\lambda$ satisfies the equation

  $$\int_{\lambda}^{\infty} \frac{e^{-\frac{1}{4} u^{2}}}{u^{2}} d u=\frac{2}{S \lambda^{3}} e^{-\frac{1}{4} \lambda^{2}}$$

  and $S=L / C_{p} \Delta \theta$.

  Integrate the left hand side of this equation by parts, to give

  $$\frac{\sqrt{\pi}}{2} \lambda e^{\frac{1}{4} \lambda^{2}} \operatorname{erfc}\left(\frac{1}{2} \lambda\right)=1-\frac{2}{S \lambda^{2}} .$$

  Hence show that a solution with $\lambda$ small must have $\lambda \approx(2 / S)^{\frac{1}{2}}$, which is self-consistent if $S$ is large.

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• A4.19

  Transport Processes

  Part II, 2003

  A shallow layer of fluid of viscosity $\mu$, density $\rho$ and depth $h(x, t)$ lies on a rigid horizontal plane $y=0$ and is bounded by impermeable barriers at $x=-L$ and $x=L$ $(L \gg h)$. Gravity acts vertically and a wind above the layer causes a shear stress $\tau(x)$ to be exerted on the upper surface in the $+x$ direction. Surface tension is negligible compared to gravity.

  (a) Assuming that the steady flow in the layer can be analysed using lubrication theory, show that the horizontal pressure gradient $p_{x}$ is given by $p_{x}=\rho g h_{x}$ and hence that

  $$h h_{x}=\frac{3}{2} \frac{\tau}{\rho g} .$$

  Show also that the fluid velocity at the surface $y=h$ is equal to $\tau h / 4 \mu$, and sketch the velocity profile for $0 \leqslant y \leqslant h$.

  (b) In the case in which $\tau$ is a constant, $\tau_{0}$, and assuming that the difference between $h$ and its average value $h_{0}$ remains small compared with $h_{0}$, show that

  $$h \approx h_{0}\left(1+\frac{3 \tau_{0} x}{2 \rho g h_{0}^{2}}\right)$$

  provided that

  $$\frac{\tau_{0} L}{\rho g h_{0}^{2}} \ll 1$$

  (c) Surfactant at surface concentration $\Gamma(x)$ is added to the surface, so that now

  $$\tau=\tau_{0}-A \Gamma_{x},$$

  where $A$ is a positive constant. The surfactant is advected by the surface fluid velocity and also experiences a surface diffusion with diffusivity $D$. Write down the equation for conservation of surfactant, and hence show that

  $$\left(\tau_{0}-A \Gamma_{x}\right) h \Gamma=4 \mu D \Gamma_{x}$$

  From equations (1), (2) and (3) deduce that

  $$\frac{\Gamma}{\Gamma_{0}}=\exp \left[\frac{\rho g}{18 \mu D}\left(h^{3}-h_{0}^{3}\right)\right]$$

  where $\Gamma_{0}$ is a constant. Assuming once more that $h_{1} \equiv h-h_{0} \ll h_{0}$, and that $h=h_{0}$ at $x=0$, show further that

  $$h_{1} \approx \frac{3 \tau_{0} x}{2 \rho g h_{0}}\left[1+\frac{A \Gamma_{0} h_{0}}{4 \mu D}\right]^{-1}$$

  provided that

  $$\frac{\tau_{0} h_{0} L}{\mu D} \ll 1 \quad \text { as well as } \quad \frac{\tau_{0} L}{\rho g h_{0}^{2}} \ll 1$$

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• A1.19

  Theoretical Geophysics

  Part II, 2003

  (i) Explain the concepts of: traction on an element of surface; the stress tensor; the strain tensor in an elastic medium. Derive a relationship between the two tensors for a linear isotropic elastic medium, stating clearly any assumption you need to make.

  (ii) State what is meant by an $\mathrm{SH}$ wave in a homogeneous isotropic elastic medium. An SH wave in a medium with shear modulus $\mu$ and density $\rho$ is incident at angle $\theta$ on an interface with a medium with shear modulus $\mu^{\prime}$ and density $\rho^{\prime}$. Evaluate the form and amplitude of the reflected wave and transmitted wave. Comment on the case $c^{\prime} \sin \theta / c>1$, where $c^{2}=\mu / \rho$ and $\left(c^{\prime}\right)^{2}=\mu^{\prime} / \rho^{\prime}$.

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• A2.16

  Theoretical Geophysics

  Part II, 2003

  (i) Explain briefly what is meant by the concepts of hydrostatic equilibrium and the buoyancy frequency. Evaluate an expression for the buoyancy frequency in an incompressible inviscid fluid with stable density profile $\rho(z)$.

  (ii) Explain briefly what is meant by the Boussinesq approximation.

  Write down the equations describing motions of small amplitude in an incompressible, stratified, Boussinesq fluid of constant buoyancy frequency.

  Derive the resulting dispersion relationship for plane wave motion. Show that there is a maximum frequency for the waves and explain briefly why this is the case.

  What would be the response to a solid body oscillating at a frequency in excess of the maximum?

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• A4.20

  Theoretical Geophysics

  Part II, 2003

  Define the Rossby number. Under what conditions will a fluid flow be at (a) high and (b) low values of the Rossby number? Briefly describe both an oceanographic and a meteorological example of each type of flow.

  Explain the concept of quasi-geostrophy for a thin layer of homogeneous fluid in a rapidly rotating system. Write down the quasi-geostrophic approximation for the vorticity in terms of the pressure, the fluid density and the rate of rotation. Define the potential vorticity and state the associated conservation law.

  A broad current flows directly eastwards ( $+x$ direction) with uniform velocity $U$ across a flat ocean basin of depth $H$. The current encounters a low, two-dimensional ridge of width $L$ and height $H h(x)(0<x<L)$, whose axis is aligned in the north-south $(y)$ direction. Neglecting any effects of stratification and assuming a constant vertical rate of rotation $\frac{1}{2} f$, such that the Rossby number is small, determine the effect of the ridge on the current. Show that the direction of the current after it leaves the ridge is dependent on the cross-sectional area of the ridge, but not on the explicit form of $h(x)$.

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• A2.17

  Mathematical Methods

  Part II, 2003

  (i) Explain how to solve the Fredholm integral equation of the second kind,

  $$f(x)=\mu \int_{a}^{b} K(x, t) f(t) d t+g(x)$$

  in the case where $K(x, t)$ is of the separable (degenerate) form

  $$K(x, t)=a_{1}(x) b_{1}(t)+a_{2}(x) b_{2}(t)$$

  (ii) For what values of the real constants $\lambda$ and $A$ does the equation

  $$u(x)=\lambda \sin x+A \int_{0}^{\pi}(\cos x \cos t+\cos 2 x \cos 2 t) u(t) d t$$

  have (a) a unique solution, (b) no solution?

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• A3.17

  Mathematical Methods

  Part II, 2003

  (i) Explain what is meant by the assertion: "the series $\sum_{0}^{\infty} b_{n} x^{n}$ is asymptotic to $f(x)$ as $x \rightarrow 0 "$.

  Consider the integral

  $$I(\lambda)=\int_{0}^{A} e^{-\lambda x} g(x) d x$$

  where $A>0, \lambda$ is real and $g$ has the asymptotic expansion

  $$g(x) \sim a_{0} x^{\alpha}+a_{1} x^{\alpha+1}+a_{2} x^{\alpha+2}+\ldots$$

  as $x \rightarrow+0$, with $\alpha>-1$. State Watson's lemma describing the asymptotic behaviour of $I(\lambda)$ as $\lambda \rightarrow \infty$, and determine an expression for the general term in the asymptotic series.

  (ii) Let

  $$h(t)=\pi^{-1 / 2} \int_{0}^{\infty} \frac{e^{-x}}{x^{1 / 2}(1+2 x t)} d x$$

  for $t \geqslant 0$. Show that

  $$h(t) \sim \sum_{k=0}^{\infty}(-1)^{k} 1.3 . \cdots \cdot(2 k-1) t^{k}$$

  as $t \rightarrow+0$.

  Suggest, for the case that $t$ is smaller than unity, the point at which this asymptotic series should be truncated so as to produce optimal numerical accuracy.

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• A4.21

  Mathematical Methods

  Part II, 2003

  Let $y(x, \lambda)$ denote the solution for $0 \leqslant x<\infty$ of

  $$\frac{d^{2} y}{d x^{2}}-\left(x+\lambda^{2}\right) y=0$$

  subject to the conditions that $y(0, \lambda)=a$ and $y(x, \lambda) \rightarrow 0$ as $x \rightarrow \infty$, where $a>0$; it may be assumed that $y(x, \lambda)>0$ for $x>0$. Write $y(x, \lambda)$ in the form

  $$y(x, \lambda)=\exp (z(x, \lambda))$$

  and consider an asymptotic expansion of the form

  $$z(x, \lambda) \sim \sum_{n=0}^{\infty} \lambda^{1-n} \phi_{n}(x),$$

  valid in the limit $\lambda \rightarrow \infty$ with $x=O(1)$. Find $\phi_{0}(x), \phi_{1}(x), \phi_{2}(x)$ and $\phi_{3}(x)$.

  It is known that the solution $y(x, \lambda)$ is of the form

  $$y(x, \lambda)=c Y(X)$$

  where

  $$X=x+\lambda^{2}$$

  and the constant factor $c$ depends on $\lambda$. By letting $Y(X)=\exp (Z(X))$, show that the expression

  $$Z(X)=-\frac{2}{3} X^{3 / 2}-\frac{1}{4} \ln X$$

  satisfies the relevant differential equation with an error of $O\left(1 / X^{3 / 2}\right)$ as $X \rightarrow \infty$. Comment on the relationship between your answers for $z(x, \lambda)$ and $Z(X)$.

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• A2.18

  Nonlinear Waves and Integrable Systems

  Part II, 2003

  (i) Write down the shock condition associated with the equation

  $$\rho_{t}+q_{x}=0$$

  where $q=q(\rho)$. Discuss briefly two possible heuristic approaches to justifying this shock condition.

  (ii) According to shallow water theory, waves on a uniformly sloping beach are described by the equations

  $$\begin{aligned} &\frac{\partial h}{\partial t}+\frac{\partial}{\partial x}(h u)=0, \\ &\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+g \frac{\partial \eta}{\partial x}=0, \quad h=\alpha x+\eta, \end{aligned}$$

  where $\alpha$ is the constant slope of the beach, $g$ is the gravitational acceleration, $u(x, t)$ is the fluid velocity, and $\eta(x, t)$ is the elevation of the fluid surface above the undisturbed level.

  Find the characteristic velocities and the characteristic form of the equations.

  What are the Riemann variables and how do they vary with $t$ on the characteristics?

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• A3.18

  Nonlinear Waves and Integrable Systems

  Part II, 2003

  (i) Write down a Lax pair for the equation

  $$i q_{t}+q_{x x}=0 .$$

  Discuss briefly, without giving mathematical details, how this pair can be used to solve the Cauchy problem on the infinite line for this equation. Discuss how this approach can be used to solve the analogous problem for the nonlinear Schrödinger equation.

  (ii) Let $q(\zeta, \eta), \tilde{q}(\zeta, \eta)$ satisfy the equations

  $$\begin{aligned} &\tilde{q}_{\zeta}=q_{\zeta}+2 \lambda \sin \frac{\tilde{q}+q}{2} \\ &\tilde{q}_{\eta}=-q_{\eta}+\frac{2}{\lambda} \sin \frac{\tilde{q}-q}{2} \end{aligned}$$

  where $\lambda$ is a constant.

  (a) Show that the above equations are compatible provided that $q, \tilde{q}$ both satisfy the Sine-Gordon equation

  $$q_{\zeta \eta}=\sin q$$

  (b) Use the above result together with the fact that

  $$\int \frac{d x}{\sin x}=\ln \left(\tan \frac{x}{2}\right)+\text { constant }$$

  to show that the one-soliton solution of the Sine-Gordon equation is given by

  $$\tan \frac{q}{4}=c \exp \left(\lambda \zeta+\frac{\eta}{\lambda}\right)$$

  where $c$ is a constant.

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• A4.22

  Nonlinear Waves and Integrable Systems

  Part II, 2003

  Let $\Phi^{+}(t), \Phi^{-}(t)$ denote the boundary values of functions which are analytic inside and outside a disc of radius $\frac{1}{2}$ centred at the origin. Let $C$ denote the boundary of this disc.

  Suppose that $\Phi^{+}, \Phi^{-}$satisfy the jump condition

  $$\Phi^{+}(t)=\frac{t}{t^{2}-1} \Phi^{-}(t)+\frac{t^{3}-t^{2}+1}{t^{2}-t}, \quad t \in C .$$

  (a) Show that the associated index is 1 .

  (b) Find the canonical solution of the homogeneous problem, i.e. the solution satisfying

  $$X(z) \sim z^{-1}, \quad z \rightarrow \infty$$

  (c) Find the general solution of the Riemann-Hilbert problem satisfying the above jump condition as well as

  $$\Phi(z)=O\left(z^{-1}\right), \quad z \rightarrow \infty$$

  (d) Use the above result to solve the linear singular integral problem

  $$\left(t^{2}+t-1\right) \phi(t)+\frac{t^{2}-t-1}{\pi i} \oint_{C} \frac{\phi(\tau)}{\tau-t} d \tau=\frac{2\left(t^{3}-t^{2}+1\right)(t+1)}{t}, \quad t \in C .$$

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• A1.1 B1.1

  Markov Chains

  Part II, 2003

  (i) Let $\left(X_{n}, Y_{n}\right)_{n \geqslant 0}$ be a simple symmetric random walk in $\mathbb{Z}^{2}$, starting from $(0,0)$, and set $T=\inf \left\{n \geqslant 0: \max \left\{\left|X_{n}\right|,\left|Y_{n}\right|\right\}=2\right\}$. Determine the quantities $\mathbb{E}(T)$ and $\mathbb{P}\left(X_{T}=2\right.$ and $\left.Y_{T}=0\right)$.

  (ii) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a discrete-time Markov chain with state-space $I$ and transition matrix $P$. What does it mean to say that a state $i \in I$ is recurrent? Prove that $i$ is recurrent if and only if $\sum_{n=0}^{\infty} p_{i i}^{(n)}=\infty$, where $p_{i i}^{(n)}$ denotes the $(i, i)$ entry in $P^{n}$.

  Show that the simple symmetric random walk in $\mathbb{Z}^{2}$ is recurrent.

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• A2.1

  Markov Chains

  Part II, 2003

  (i) What is meant by a Poisson process of rate $\lambda$ ? Show that if $\left(X_{t}\right)_{t \geqslant 0}$ and $\left(Y_{t}\right)_{t \geqslant 0}$ are independent Poisson processes of rates $\lambda$ and $\mu$ respectively, then $\left(X_{t}+Y_{t}\right)_{t \geqslant 0}$ is also a Poisson process, and determine its rate.

  (ii) A Poisson process of rate $\lambda$ is observed by someone who believes that the first holding time is longer than all subsequent holding times. How long on average will it take before the observer is proved wrong?

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• A3.1 B3.1

  Markov Chains

  Part II, 2003

  (i) Consider the continuous-time Markov chain $\left(X_{t}\right)_{t \geqslant 0}$ with state-space $\{1,2,3,4\}$ and $Q$-matrix

  $$Q=\left(\begin{array}{rrrr} -2 & 0 & 0 & 2 \\ 1 & -3 & 2 & 0 \\ 0 & 2 & -2 & 0 \\ 1 & 5 & 2 & -8 \end{array}\right)$$

  Set

  $$Y_{t}= \begin{cases}X_{t} & \text { if } X_{t} \in\{1,2,3\} \\ 2 & \text { if } X_{t}=4\end{cases}$$

  and

  $$Z_{t}= \begin{cases}X_{t} & \text { if } X_{t} \in\{1,2,3\} \\ 1 & \text { if } X_{t}=4\end{cases}$$

  Determine which, if any, of the processes $\left(Y_{t}\right)_{t \geqslant 0}$ and $\left(Z_{t}\right)_{t \geqslant 0}$ are Markov chains.

  (ii) Find an invariant distribution for the chain $\left(X_{t}\right)_{t \geqslant 0}$ given in Part (i). Suppose $X_{0}=1$. Find, for all $t \geqslant 0$, the probability that $X_{t}=1$.

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• A4.1

  Markov Chains

  Part II, 2003

  Consider a pack of cards labelled $1, \ldots, 52$. We repeatedly take the top card and insert it uniformly at random in one of the 52 possible places, that is, either on the top or on the bottom or in one of the 50 places inside the pack. How long on average will it take for the bottom card to reach the top?

  Let $p_{n}$ denote the probability that after $n$ iterations the cards are found in increasing order. Show that, irrespective of the initial ordering, $p_{n}$ converges as $n \rightarrow \infty$, and determine the limit $p$. You should give precise statements of any general results to which you appeal.

  Show that, at least until the bottom card reaches the top, the ordering of cards inserted beneath it is uniformly random. Hence or otherwise show that, for all $n$,

  $$\left|p_{n}-p\right| \leqslant 52(1+\log 52) / n$$

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• A1.2 B1.2

  Principles of Dynamics

  Part II, 2003

  (i) Consider $N$ particles moving in 3 dimensions. The Cartesian coordinates of these particles are $x^{A}(t), A=1, \ldots, 3 N$. Now consider an invertible change of coordinates to coordinates $q^{a}\left(x^{A}, t\right), \quad a=1, \ldots, 3 N$, so that one may express $x^{A}$ as $x^{A}\left(q^{a}, t\right)$. Show that the velocity of the system in Cartesian coordinates $\dot{x}^{A}(t)$ is given by the following expression:

  $$\dot{x}^{A}\left(\dot{q}^{a}, q^{a}, t\right)=\sum_{b=1}^{3 N} \dot{q}^{b} \frac{\partial x^{A}}{\partial q^{b}}\left(q^{a}, t\right)+\frac{\partial x^{A}}{\partial t}\left(q^{a}, t\right)$$

  Furthermore, show that Lagrange's equations in the two coordinate systems are related via

  $$\frac{\partial L}{\partial q^{a}}-\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}^{a}}\right)=\sum_{A=1}^{3 N} \frac{\partial x^{A}}{\partial q^{a}}\left(\frac{\partial L}{\partial x^{A}}-\frac{d}{d t} \frac{\partial L}{\partial \dot{x}^{A}}\right)$$

  (ii) Now consider the case where there are $p<3 N$ constraints applied, $f^{\ell}\left(x^{A}, t\right)=$ $0, \ell=1, \ldots, p$. By considering the $f^{\ell}, \ell=1, \ldots, p$, and a set of independent coordinates $q^{a}, a=1, \ldots, 3 N-p$, as a set of $3 N$ new coordinates, show that the Lagrange equations of the constrained system, i.e.

  $$\begin{gathered} \frac{\partial L}{\partial x^{A}}-\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{x}^{A}}\right)+\sum_{\ell=1}^{p} \lambda^{\ell} \frac{\partial f^{\ell}}{\partial x^{A}}=0, \quad A=1, \ldots, 3 N \\ f^{\ell}=0, \quad \ell=1, \ldots, p \end{gathered}$$

  (where the $\lambda^{\ell}$ are Lagrange multipliers) imply Lagrange's equations for the unconstrained coordinates, i.e.

  $$\frac{\partial L}{\partial q^{a}}-\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}^{a}}\right)=0, \quad a=1, \ldots, 3 N-p .$$

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• A2.2 B2.1

  Principles of Dynamics

  Part II, 2003

  (i) The trajectory $\mathbf{x}(t)$ of a non-relativistic particle of mass $m$ and charge $q$ moving in an electromagnetic field obeys the Lorentz equation

  $$m \ddot{\mathbf{x}}=q\left(\mathbf{E}+\frac{\dot{\mathbf{x}}}{c} \wedge \mathbf{B}\right) .$$

  Show that this equation follows from the Lagrangian

  $$L=\frac{1}{2} m \dot{\mathbf{x}}^{2}-q\left(\phi-\frac{\dot{\mathbf{x}} \cdot \mathbf{A}}{c}\right)$$

  where $\phi(\mathbf{x}, t)$ is the electromagnetic scalar potential and $\mathbf{A}(\mathbf{x}, t)$ the vector potential, so that

  $$\mathbf{E}=-\frac{1}{c} \dot{\mathbf{A}}-\nabla \phi \text { and } \mathbf{B}=\nabla \wedge \mathbf{A}$$

  (ii) Let $\mathbf{E}=0$. Consider a particle moving in a constant magnetic field which points in the $z$ direction. Show that the particle moves in a helix about an axis pointing in the $z$ direction. Evaluate the radius of the helix.

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• A3.2

  Principles of Dynamics

  Part II, 2003

  (i) An axisymmetric bowling ball of mass $M$ has the shape of a sphere of radius $a$. However, it is biased so that the centre of mass is located a distance $a / 2$ away from the centre, along the symmetry axis.

  The three principal moments of inertia about the centre of mass are $(A, A, C)$. The ball starts out in a stable equilibrium at rest on a perfectly frictionless flat surface with the symmetry axis vertical. The symmetry axis is then tilted through $\theta_{0}$, the ball is spun about this axis with an angular velocity $n$, and the ball is released.

  Explain why the centre of mass of the ball moves only in the vertical direction during the subsequent motion. Write down the Lagrangian for the ball in terms of the usual Euler angles $\theta, \phi$ and $\psi$.

  (ii) Show that there are three independent constants of the motion. Eliminate two of the angles from the Lagrangian and find the effective Lagrangian for the coordinate $\theta$.

  Find the maximum and minimum values of $\theta$ in the motion of the ball when the quantity $\frac{C^{2} n^{2}}{A M g a}$ is (a) very small and (b) very large.

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• A4.2

  Principles of Dynamics

  Part II, 2003

  The action $S$ of a Hamiltonian system may be regarded as a function of the final coordinates $q^{a}, a=1, \ldots, N$, and the final time $t$ by setting

  $$S\left(q^{a}, t\right)=\int_{\left(q_{i}^{a}, t_{i}\right)}^{\left(q^{a}, t\right)} d t^{\prime}\left[p^{a}\left(t^{\prime}\right) \dot{q}^{a}\left(t^{\prime}\right)-H\left(p^{a}\left(t^{\prime}\right), q^{a}\left(t^{\prime}\right), t^{\prime}\right)\right]$$

  where the initial coordinates $q_{i}^{a}$ and time $t_{i}$ are held fixed, and $p^{a}\left(t^{\prime}\right), q^{a}\left(t^{\prime}\right)$ are the solutions to Hamilton's equations with Hamiltonian $H$, satisfying $q^{a}(t)=q^{a}, q^{a}\left(t_{i}\right)=q_{i}^{a}$.

  (a) Show that under an infinitesimal change of the final coordinates $\delta q^{a}$ and time $\delta t$, the change in $S$ is

  $$\delta S=p_{a}(t) \delta q_{a}-H\left(p^{a}(t), q^{a}(t), t\right) \delta t$$

  (b) Hence derive the Hamilton-Jacobi equation

  $$\frac{\partial S}{\partial t}\left(q^{a}, t\right)+H\left(\frac{\partial S}{\partial q^{a}}\left(q^{a}, t\right), q^{a}, t\right)=0$$

  (c) If we can find a solution to $(*)$,

  $$S=S\left(q^{a}, t ; P^{a}\right),$$

  where $P^{a}$ are $N$ integration constants, then we can use $S$ as a generating function of type $I I$, where

  $$p^{a}=\frac{\partial S}{\partial q^{a}} \quad, \quad Q^{a}=-\frac{\partial S}{\partial P^{a}}$$

  Show that the Hamiltonian $K$ in the new coordinates $Q^{a}, P^{a}$ vanishes.

  (d) Write down and solve the Hamilton-Jacobi equation for the one-dimensional simple harmonic oscillator, where $H=\frac{1}{2}\left(p^{2}+q^{2}\right)$. Show the solution takes the form $S(q, t ; E)=W(q, E)-E t$. Using this as a generating function $F_{I I}(q, t, P)$ show that the new coordinates $Q, P$ are constants of the motion and give their physical interpretation.

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• A1.3

  Functional Analysis

  Part II, 2003

  (i) Let $T: H_{1} \rightarrow H_{2}$ be a continuous linear map between two Hilbert spaces $H_{1}, H_{2}$. Define the adjoint $T^{*}$ of $T$. Explain what it means to say that $T$ is Hermitian or unitary.

  Let $\phi: \mathbb{R} \rightarrow \mathbb{C}$ be a bounded continuous function. Show that the map

  $$T: L^{2}(\mathbb{R}) \rightarrow L^{2}(\mathbb{R})$$

  with $T f(x)=\phi(x) f(x+1)$ is a continuous linear map and find its adjoint. When is $T$ Hermitian? When is it unitary?

  (ii) Let $C$ be a closed, non-empty, convex subset of a real Hilbert space $H$. Show that there exists a unique point $x_{o} \in C$ with minimal norm. Show that $x_{o}$ is characterised by the property

  $$\left\langle x_{o}-x, x_{o}\right\rangle \leqslant 0 \quad \text { for all } x \in C .$$

  Does this result still hold when $C$ is not closed or when $C$ is not convex? Justify your answers.

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• A2.3 B2.2

  Functional Analysis

  Part II, 2003

  (i) Define the dual of a normed vector space $(E,\|\cdot\|)$. Show that the dual is always a complete normed space.

  Prove that the vector space $\ell_{1}$, consisting of those real sequences $\left(x_{n}\right)_{n=1}^{\infty}$ for which the norm

  $$\left\|\left(x_{n}\right)\right\|_{1}=\sum_{n=1}^{\infty}\left|x_{n}\right|$$

  is finite, has the vector space $\ell_{\infty}$ of all bounded sequences as its dual.

  (ii) State the Stone-Weierstrass approximation theorem.

  Let $K$ be a compact subset of $\mathbb{R}^{n}$. Show that every $f \in C_{\mathbb{R}}(K)$ can be uniformly approximated by a sequence of polynomials in $n$ variables.

  Let $f$ be a continuous function on $[0,1] \times[0,1]$. Deduce that

  $$\int_{0}^{1}\left(\int_{0}^{1} f(x, y) d x\right) d y=\int_{0}^{1}\left(\int_{0}^{1} f(x, y) d y\right) d x$$

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• A3.3 B3.2

  Functional Analysis

  Part II, 2003

  (i) Let $p$ be a point of the compact interval $I=[a, b] \subset \mathbb{R}$ and let $\delta_{p}: C(I) \rightarrow \mathbb{R}$ be defined by $\delta_{p}(f)=f(p)$. Show that

  $$\delta_{p}:\left(C(I),\|\cdot\|_{\infty}\right) \rightarrow \mathbb{R}$$

  is a continuous, linear map but that

  $$\delta_{p}:\left(C(I),\|\cdot\|_{1}\right) \rightarrow \mathbb{R}$$

  is not continuous.

  (ii) Consider the space $C^{(n)}(I)$ of $n$-times continuously differentiable functions on the interval $I$. Write

  $$\|f\|_{\infty}^{(n)}=\sum_{k=0}^{n}\left\|f^{(k)}\right\|_{\infty} \quad \text { and } \quad\|f\|_{1}^{(n)}=\sum_{r=0}^{n}\left\|f^{(k)}\right\|_{1}$$

  for $f \in C^{(n)}(I)$. Show that $\left(C^{(n)}(I),\left\|^{\prime} \cdot\right\|_{\infty}^{(n)}\right)$ is a complete normed space. Is the space $\left(C^{(n)}(I),\|\cdot\|_{1}^{(n)}\right)$ also complete?

  Let $f: I \rightarrow I$ be an $n$-times continuously differentiable map and define

  $$\mu_{f}: C^{(n)}(I) \rightarrow C^{(n)}(I) \quad \text { by } \quad g \mapsto g \circ f .$$

  Show that $\mu_{f}$ is a continuous linear map when $C^{(n)}(I)$ is equipped with the norm $\|\cdot\|_{\infty}^{(n)}$.

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• A4.3

  Functional Analysis

  Part II, 2003

  (i) State the Monotone Convergence Theorem and explain briefly how to prove it.

  (ii) For which real values of $\alpha$ is $x^{-\alpha} \log x \in L^{1}((1, \infty))$ ?

  Let $p>0$. Using the Monotone Convergence Theorem and the identity

  $$\frac{1}{x^{p}(x-1)}=\sum_{n=0}^{\infty} \frac{1}{x^{p+n+1}}$$

  prove carefully that

  $$\int_{1}^{\infty} \frac{\log x}{x^{p}(x-1)} d x=\sum_{n=0}^{\infty} \frac{1}{(n+p)^{2}}$$

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• A1.4

  Groups, Rings and Fields

  Part II, 2003

  (i) Let $p$ be a prime number. Show that a group $G$ of order $p^{n}(n \geqslant 2)$ has a nontrivial normal subgroup, that is, $G$ is not a simple group.

  (ii) Let $p$ and $q$ be primes, $p>q$. Show that a group $G$ of order $p q$ has a normal Sylow $p$-subgroup. If $G$ has also a normal Sylow $q$-subgroup, show that $G$ is cyclic. Give a necessary and sufficient condition on $p$ and $q$ for the existence of a non-abelian group of order $p q$. Justify your answer.

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• B1.3

  Groups, Rings and Fields

  Part II, 2003

  (i) Let $p$ be a prime number. Show that a group $G$ of order $p^{n}(n \geqslant 2)$ has a nontrivial normal subgroup, that is, $G$ is not a simple group.

  (ii) Let $p$ and $q$ be primes, $p>q$. Show that a group $G$ of order $p q$ has a normal Sylow $p$-subgroup. If $G$ has also a normal Sylow $q$-subgroup, show that $G$ is cyclic. Give a necessary and sufficient condition on $p$ and $q$ for the existence of a non-abelian group of order $p q$. Justify your answer.

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• A3.4

  Groups, Rings and Fields

  Part II, 2003

  (i) Let $K$ be the splitting field of the polynomial $f=X^{3}-2$ over the rationals. Find the Galois group $G$ of $K / \mathbb{Q}$ and describe its action on the roots of $f$.

  (ii) Let $K$ be the splitting field of the polynomial $X^{4}+a X^{2}+b$ (where $a, b \in \mathbb{Q}$ ) over the rationals. Assuming that the polynomial is irreducible, prove that the Galois group $G$ of the extension $K / \mathbb{Q}$ is either $C_{4}$, or $C_{2} \times C_{2}$, or the dihedral group $D_{8}$.

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• A2.4 B2.3

  Groups, Rings and Fields

  Part II, 2003

  (i) In each of the following two cases, determine a highest common factor in $\mathbb{Z}[i]$ :

  (a) $3+4 i, 4-3 i$;

  (b) $3+4 i, 1+2 i$.

  (ii) State and prove the Eisenstein criterion for irreducibility of polynomials with integer coefficients. Show that, if $p$ is prime, the polynomial

  $$1+x+\cdots+x^{p-1}$$

  is irreducible over $\mathbb{Z}$.

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• A4.4

  Groups, Rings and Fields

  Part II, 2003

  Write an essay on the theory of invariants. Your essay should discuss the theorem on the finite generation of the ring of invariants, the theorem on elementary symmetric functions, and some examples of calculation of rings of invariants.

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• A $1 . 5 \quad$ B $1 . 4 \quad$

  Electromagnetism

  Part II, 2003

  (i) Using Maxwell's equations as they apply to magnetostatics, show that the magnetic field $\mathbf{B}$ can be described in terms of a vector potential $\mathbf{A}$ on which the condition $\nabla \cdot \mathbf{A}=0$ may be imposed. Hence derive an expression, valid at any point in space, for the vector potential due to a steady current distribution of density $\mathbf{j}$ that is non-zero only within a finite domain.

  (ii) Verify that the vector potential $\mathbf{A}$ that you found in Part (i) satisfies $\nabla \cdot \mathbf{A}=0$, and use it to obtain the Biot-Savart law expression for $\mathbf{B}$. What is the corresponding result for a steady surface current distribution of density $\mathbf{s}$ ?

  In cylindrical polar coordinates $(\rho, \phi, z)$ (oriented so that $\mathbf{e}_{\rho} \times \mathbf{e}_{\phi}=\mathbf{e}_{z}$ ) a surface current

  $$\mathbf{s}=s(\rho) \mathbf{e}_{\phi}$$

  flows in the plane $z=0$. Given that

  $$s(\rho)= \begin{cases}4 I\left(1+\frac{a^{2}}{\rho^{2}}\right)^{\frac{1}{2}} & a \leqslant \rho \leqslant 3 a \\ 0 & \text { otherwise }\end{cases}$$

  show that the magnetic field at the point $\mathbf{r}=a \mathbf{e}_{z}$ has $z$-component

  $$B_{z}=\mu_{0} I \log 5 .$$

  State, with justification, the full result for $\mathbf{B}$ at the point $\mathbf{r}=a \mathbf{e}_{z}$.

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• A2 $. 5 \quad$

  Electromagnetism

  Part II, 2003

  (i) A plane electromagnetic wave has electric and magnetic fields

  $$\mathbf{E}=\mathbf{E}_{0} e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)}, \quad \mathbf{B}=\mathbf{B}_{0} e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)}$$

  for constant vectors $\mathbf{E}_{0}, \mathbf{B}_{0}$, constant positive angular frequency $\omega$ and constant wavevector $\mathbf{k}$. Write down the vacuum Maxwell equations and show that they imply

  $$\mathbf{k} \cdot \mathbf{E}_{0}=0, \quad \mathbf{k} \cdot \mathbf{B}_{0}=0, \quad \omega \mathbf{B}_{0}=\mathbf{k} \times \mathbf{E}_{0}$$

  Show also that $|\mathbf{k}|=\omega / c$, where $c$ is the speed of light.

  (ii) State the boundary conditions on $\mathbf{E}$ and $\mathbf{B}$ at the surface $S$ of a perfect conductor. Let $\sigma$ be the surface charge density and s the surface current density on $S$. How are $\sigma$ and $\mathbf{s}$ related to $\mathbf{E}$ and $\mathbf{B}$ ?

  A plane electromagnetic wave is incident from the half-space $x<0$ upon the surface $x=0$ of a perfectly conducting medium in $x>0$. Given that the electric and magnetic fields of the incident wave take the form $(*)$ with

  $$\mathbf{k}=k(\cos \theta, \sin \theta, 0) \quad(0<\theta<\pi / 2)$$

  and

  $$\mathbf{E}_{0}=\lambda(-\sin \theta, \cos \theta, 0),$$

  find $\mathbf{B}_{0}$.

  Reflection of the incident wave at $x=0$ produces a reflected wave with electric field

  $$\mathbf{E}_{0}^{\prime} e^{i\left(\mathbf{k}^{\prime} \cdot \mathbf{r}-\omega t\right)}$$

  with

  $$\mathbf{k}^{\prime}=k(-\cos \theta, \sin \theta, 0)$$

  By considering the boundary conditions at $x=0$ on the total electric field, show that

  $$\mathbf{E}_{0}^{\prime}=-\lambda(\sin \theta, \cos \theta, 0)$$

  Show further that the electric charge density on the surface $x=0$ takes the form

  $$\sigma=\sigma_{0} e^{i k(y \sin \theta-c t)}$$

  for a constant $\sigma_{0}$ that you should determine. Find the magnetic field of the reflected wave and hence the surface current density $\mathbf{s}$ on the surface $x=0$.

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• A3.5 B3.3

  Electromagnetism

  Part II, 2003

  (i) Given the electric field (in cartesian components)

  $$\mathbf{E}(\mathbf{r}, t)=\left(0, x / t^{2}, 0\right)$$

  use the Maxwell equation

  $$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$$

  to find $\mathbf{B}$ subject to the boundary condition that $|\mathbf{B}| \rightarrow 0$ as $t \rightarrow \infty$.

  Let $S$ be the planar rectangular surface in the $x y$-plane with corners at

  $$(0,0,0), \quad(L, 0,0), \quad(L, a, 0), \quad(0, a, 0)$$

  where $a$ is a constant and $L=L(t)$ is some function of time. The magnetic flux through $S$ is given by the surface integral

  $$\Phi=\int_{S} \mathbf{B} \cdot d \mathbf{S}$$

  Compute $\Phi$ as a function of $t$.

  Let $\mathcal{C}$ be the closed rectangular curve that bounds the surface $S$, taken anticlockwise in the $x y$-plane, and let $\mathbf{v}$ be its velocity (which depends, in this case, on the segment of $\mathcal{C}$ being considered). Compute the line integral

  $$\oint_{\mathcal{C}}(\mathbf{E}+\mathbf{v} \times \mathbf{B}) \cdot d \mathbf{r}$$

  Hence verify that

  $$\oint_{\mathcal{C}}(\mathbf{E}+\mathbf{v} \times \mathbf{B}) \cdot d \mathbf{r}=-\frac{d \Phi}{d t}$$

  (ii) A surface $S$ is bounded by a time-dependent closed curve $\mathcal{C}(t)$ such that in time $\delta t$ it sweeps out a volume $\delta V$. By considering the volume integral

  $$\int_{\delta V} \nabla \cdot \mathbf{B} d \tau$$

  and using the divergence theorem, show that the Maxwell equation $\nabla \cdot \mathbf{B}=0$ implies that

  $$\frac{d \Phi}{d t}=\int_{S} \frac{\partial \mathbf{B}}{\partial t} \cdot d \mathbf{S}-\oint_{\mathcal{C}}(\mathbf{v} \times \mathbf{B}) \cdot d \mathbf{r}$$

  where $\Phi$ is the magnetic flux through $S$ as given in Part (i). Hence show, using (1) and Stokes' theorem, that (2) is a consequence of Maxwell's equations.

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• A4.5

  Electromagnetism

  Part II, 2003

  Let $\mathbf{E}(\mathbf{r})$ be the electric field due to a continuous static charge distribution $\rho(\mathbf{r})$ for which $|\mathbf{E}| \rightarrow 0$ as $|\mathbf{r}| \rightarrow \infty$. Starting from consideration of a finite system of point charges, deduce that the electrostatic energy of the charge distribution $\rho$ is

  $$W=\frac{1}{2} \varepsilon_{0} \int|\mathbf{E}|^{2} d \tau$$

  where the volume integral is taken over all space.

  A sheet of perfectly conducting material in the form of a surface $S$, with unit normal $\mathbf{n}$, carries a surface charge density $\sigma$. Let $E_{\pm}=\mathbf{n} \cdot \mathbf{E}_{\pm}$denote the normal components of the electric field $\mathbf{E}$ on either side of $S$. Show that

  $$\frac{1}{\varepsilon_{0}} \sigma=E_{+}-E_{-} .$$

  Three concentric spherical shells of perfectly conducting material have radii $a, b, c$ with $a<b<c$. The innermost and outermost shells are held at zero electric potential. The other shell is held at potential $V$. Find the potentials $\phi_{1}(r)$ in $a<r<b$ and $\phi_{2}(r)$ in $b<r<c$. Compute the surface charge density $\sigma$ on the shell of radius $b$. Use the formula $(*)$ to compute the electrostatic energy of the system.

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• A1.6

  Dynamics of Differential Equations

  Part II, 2003

  (i) State and prove Dulac's Criterion for the non-existence of periodic orbits in $\mathbb{R}^{2}$. Hence show (choosing a weighting factor of the form $x^{\alpha} y^{\beta}$ ) that there are no periodic orbits of the equations

  $$\dot{x}=x\left(2-6 x^{2}-5 y^{2}\right), \quad \dot{y}=y\left(-3+10 x^{2}+3 y^{2}\right)$$

  (ii) State the Poincaré-Bendixson Theorem. A model of a chemical reaction (the Brusselator) is defined by the second order system

  $$\dot{x}=a-x(1+b)+x^{2} y, \quad \dot{y}=b x-x^{2} y$$

  where $a, b$ are positive parameters. Show that there is a unique fixed point. Show that, for a suitable choice of $p>0$, trajectories enter the closed region bounded by $x=p, y=b / p$, $x+y=a+b / p$ and $y=0$. Deduce that when $b>1+a^{2}$, the system has a periodic orbit.

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• A2.6 B2.4

  Dynamics of Differential Equations

  Part II, 2003

  (i) What is a Liapunov function?

  Consider the second order ODE

  $$\dot{x}=y, \quad \dot{y}=-y-\sin ^{3} x$$

  By finding a suitable Liapunov function of the form $V(x, y)=f(x)+g(y)$, where $f$ and $g$ are to be determined, show that the origin is asymptotically stable. Using your form of $V$, find the greatest value of $y_{0}$ such that a trajectory through $\left(0, y_{0}\right)$ is guaranteed to tend to the origin as $t \rightarrow \infty$.

  [Any theorems you use need not be proved but should be clearly stated.]

  (ii) Explain the use of the stroboscopic method for investigating the dynamics of equations of the form $\ddot{x}+x=\epsilon f(x, \dot{x}, t)$, when $|\epsilon| \ll 1$. In particular, for $x=R \cos (t+\theta)$, $\dot{x}=-R \sin (t+\theta)$ derive the equations, correct to order $\epsilon$,

  $$\dot{R}=-\epsilon\langle f \sin (t+\theta)\rangle, \quad R \dot{\theta}=-\epsilon\langle f \cos (t+\theta)\rangle$$

  where the brackets denote an average over the period of the unperturbed oscillator.

  Find the form of the right hand sides of these equations explicitly when $f=$ $\Gamma x^{2} \cos t-3 q x$, where $\Gamma>0, q \neq 0$. Show that apart from the origin there is another fixed point of $(*)$, and determine its stability. Sketch the trajectories in $(R, \theta)$ space in the case $q>0$. What do you deduce about the dynamics of the full equation?

  [You may assume that $\left\langle\cos ^{2} t\right\rangle=\frac{1}{2},\left\langle\cos ^{4} t\right\rangle=\frac{3}{8},\left\langle\cos ^{2} t \sin ^{2} t\right\rangle=\frac{1}{8}$.]

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• A3.6 B3.4

  Dynamics of Differential Equations

  Part II, 2003

  (i) Define the Poincaré index of a curve $\mathcal{C}$ for a vector field $\mathbf{f}(\mathbf{x}), \mathbf{x} \in \mathbb{R}^{2}$. Explain why the index is uniquely given by the sum of the indices for small curves around each fixed point within $\mathcal{C}$. Write down the indices for a saddle point and for a focus (spiral) or node, and show that the index of a periodic solution of $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ has index unity.

  A particular system has a periodic orbit containing five fixed points, and two further periodic orbits. Sketch the possible arrangements of these orbits, assuming there are no degeneracies.

  (ii) A dynamical system in $\mathbb{R}^{2}$ depending on a parameter $\mu$ has a homoclinic orbit when $\mu=\mu_{0}$. Explain how to determine the stability of this orbit, and sketch the different behaviours for $\mu<\mu_{0}$ and $\mu>\mu_{0}$ in the case that the orbit is stable.

  Now consider the system

  $$\dot{x}=y, \quad \dot{y}=x-x^{2}+y(\alpha+\beta x)$$

  where $\alpha, \beta$ are constants. Show that the origin is a saddle point, and that if there is an orbit homoclinic to the origin then $\alpha, \beta$ are related by

  $$\oint y^{2}(\alpha+\beta x) d t=0$$

  where the integral is taken round the orbit. Evaluate this integral for small $\alpha, \beta$ by approximating $y$ by its form when $\alpha=\beta=0$. Hence give conditions on (small) $\alpha, \beta$ that lead to a stable homoclinic orbit at the origin. [Note that $y d t=d x$.]

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• A4.6

  Dynamics of Differential Equations

  Part II, 2003

  Explain what is meant by a steady-state bifurcation of a fixed point $\mathbf{x}_{0}(\mu)$ of an ODE $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}, \mu)$, in $\mathbb{R}^{n}$, where $\mu$ is a real parameter. Give examples for $n=1$ of equations exhibiting saddle-node, transcritical and pitchfork bifurcations.

  Consider the system in $\mathbb{R}^{2}$, with $\mu>0$,

  $$\dot{x}=x\left(1-y-4 x^{2}\right), \quad \dot{y}=y\left(\mu-y-x^{2}\right) .$$

  Show that the fixed point $(0, \mu)$ has a bifurcation when $\mu=1$, while the fixed points $\left(\pm \frac{1}{2}, 0\right)$ have a bifurcation when $\mu=\frac{1}{4}$. By finding the first approximation to the extended centre manifold, construct the normal form at the bifurcation point in each case, and determine the respective bifurcation types. Deduce that for $\mu$ just greater than $\frac{1}{4}$, and for $\mu$ just less than 1 , there is a stable pair of "mixed-mode" solutions with $x^{2}>0, y>0$.

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• A $1 . 7 \quad$ B1.12

  Logic, Computation and Set Theory

  Part II, 2003

  (i) State Zorn's Lemma. Use Zorn's Lemma to prove that every real vector space has a basis.

  (ii) State the Bourbaki-Witt Theorem, and use it to prove Zorn's Lemma, making clear where in the argument you appeal to the Axiom of Choice.

  Conversely, deduce the Bourbaki-Witt Theorem from Zorn's Lemma.

  If $X$ is a non-empty poset in which every chain has an upper bound, must $X$ be chain-complete?

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• B2.11

  Logic, Computation and Set Theory

  Part II, 2003

  State the Axiom of Replacement.

  Show that for any set $x$ there is a transitive set $y$ that contains $x$, indicating where in your argument you have used the Axiom of Replacement. No form of recursion theorem may be assumed without proof.

  Which of the following are true and which are false? Give proofs or counterexamples as appropriate. You may assume standard properties of ordinals.

  (a) If $x$ is a transitive set then $x$ is an ordinal.

  (b) If each member of a set $x$ is an ordinal then $x$ is an ordinal.

  (c) If $x$ is a transitive set and each member of $x$ is an ordinal then $x$ is an ordinal.

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• A3.8 B3.11

  Logic, Computation and Set Theory

  Part II, 2003

  (i) What does it mean for a function from $\mathbb{N}^{k}$ to $\mathbb{N}$ to be recursive? Write down a function that is not recursive. You should include a proof that your example is not recursive.

  (ii) What does it mean for a subset of $\mathbb{N}^{k}$ to be recursive, and what does it mean for it to be recursively enumerable? Give, with proof, an example of a set that is recursively enumerable but not recursive. Prove that a set is recursive if and only if both it and its complement are recursively enumerable. If a set is recursively enumerable, must its complement be recursively enumerable?

  [You may assume the existence of any universal recursive functions or universal register machine programs that you wish.]

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• A4.8 B4.10

  Logic, Computation and Set Theory

  Part II, 2003

  Write an essay on propositional logic. You should include all relevant definitions, and should cover the Completeness Theorem, as well as the Compactness Theorem and the Decidability Theorem.

  [You may assume that the set of primitive propositions is countable. You do not need to give proofs of simple examples of syntactic implication, such as the fact that $p \Rightarrow p$ is a theorem or that $p \Rightarrow q$ and $q \Rightarrow r$ syntactically imply $p \Rightarrow r$.]

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• A1.12 B1.15

  Principles of Statistics

  Part II, 2003

  (i) A public health official is seeking a rational policy of vaccination against a relatively mild ailment which causes absence from work. Surveys suggest that $60 \%$ of the population are already immune, but accurate tests to detect vulnerability in any individual are too costly for mass screening. A simple skin test has been developed, but is not completely reliable. A person who is immune to the ailment will have a negligible reaction to the skin test with probability $0.4$, a moderate reaction with probability $0.5$ and a strong reaction with probability 0.1. For a person who is vulnerable to the ailment the corresponding probabilities are $0.1,0.4$ and $0.5$. It is estimated that the money-equivalent of workhours lost from failing to vaccinate a vulnerable person is 20 , that the unnecessary cost of vaccinating an immune person is 8 , and that there is no cost associated with vaccinating a vulnerable person or failing to vaccinate an immune person. On the basis of the skin test, it must be decided whether to vaccinate or not. What is the Bayes decision rule that the health official should adopt?

  (ii) A collection of $I$ students each sit $J$ exams. The ability of the $i$ th student is represented by $\theta_{i}$ and the performance of the $i$ th student on the $j$ th exam is measured by $X_{i j}$. Assume that, given $\boldsymbol{\theta}=\left(\theta_{1}, \ldots, \theta_{I}\right)$, an appropriate model is that the variables $\left\{X_{i j}, 1 \leqslant i \leqslant I, 1 \leqslant j \leqslant J\right\}$ are independent, and

  $$X_{i j} \sim N\left(\theta_{i}, \tau^{-1}\right),$$

  for a known positive constant $\tau$. It is reasonable to assume, a priori, that the $\theta_{i}$ are independent with

  $$\theta_{i} \sim N\left(\mu, \zeta^{-1}\right),$$

  where $\mu$ and $\zeta$ are population parameters, known from experience with previous cohorts of students.

  Compute the posterior distribution of $\boldsymbol{\theta}$ given the observed exam marks vector $\mathbf{X}=\left\{X_{i j}, 1 \leqslant i \leqslant I, 1 \leqslant j \leqslant J\right\} .$

  Suppose now that $\tau$ is also unknown, but assumed to have a $\operatorname{Gamma}\left(\alpha_{0}, \beta_{0}\right)$ distribution, for known $\alpha_{0}, \beta_{0}$. Compute the posterior distribution of $\tau$ given $\boldsymbol{\theta}$ and $\mathbf{X}$ Find, up to a normalisation constant, the form of the marginal density of $\boldsymbol{\theta}$ given $\mathbf{X}$.

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• A2.11 B2.16

  Principles of Statistics

  Part II, 2003

  (i) Outline briefly the Bayesian approach to hypothesis testing based on Bayes factors.

  (ii) Let $Y_{1}, Y_{2}$ be independent random variables, both uniformly distributed on $\left(\theta-\frac{1}{2}, \theta+\frac{1}{2}\right)$. Find a minimal sufficient statistic for $\theta$. Let $Y_{(1)}=\min \left\{Y_{1}, Y_{2}\right\}$, $Y_{(2)}=\max \left\{Y_{1}, Y_{2}\right\}$. Show that $R=Y_{(2)}-Y_{(1)}$ is ancilliary and explain why the Conditionality Principle would lead to inference about $\theta$ being drawn from the conditional distribution of $\frac{1}{2}\left\{Y_{(1)}+Y_{(2)}\right\}$ given $R$. Find the form of this conditional distribution.

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• A3.12 B3.15

  Principles of Statistics

  Part II, 2003

  (i) Let $X_{1}, \ldots, X_{n}$ be independent, identically distributed random variables, with the exponential density $f(x ; \theta)=\theta e^{-\theta x}, x>0$.

  Obtain the maximum likelihood estimator $\hat{\theta}$ of $\theta$. What is the asymptotic distribution of $\sqrt{n}(\hat{\theta}-\theta)$ ?

  What is the minimum variance unbiased estimator of $\theta ?$ Justify your answer carefully.

  (ii) Explain briefly what is meant by the profile log-likelihood for a scalar parameter of interest $\gamma$, in the presence of a nuisance parameter $\xi$. Describe how you would test a null hypothesis of the form $H_{0}: \gamma=\gamma_{0}$ using the profile log-likelihood ratio statistic.

  In a reliability study, lifetimes $T_{1}, \ldots, T_{n}$ are independent and exponentially distributed, with means of the form $E\left(T_{i}\right)=\exp \left(\beta+\xi z_{i}\right)$ where $\beta, \xi$ are unknown and $z_{1}, \ldots, z_{n}$ are known constants. Inference is required for the mean lifetime, $\exp \left(\beta+\xi z_{0}\right)$, for covariate value $z_{0}$.

  Find, as explicitly as possible, the profile log-likelihood for $\gamma \equiv \beta+\xi z_{0}$, with nuisance parameter $\xi$.

  Show that, under $H_{0}: \gamma=\gamma_{0}$, the profile $\log -$ likelihood ratio statistic has a distribution which does not depend on the value of $\xi$. How might the parametric bootstrap be used to obtain a test of $H_{0}$ of exact size $\alpha$ ?

  [Hint: if $Y$ is exponentially distributed with mean 1 , then $\mu Y$ is exponentially distributed with mean $\mu$.]

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• A4.13 B4.15

  Principles of Statistics

  Part II, 2003

  Write an account, with appropriate examples, of inference in multiparameter exponential families. Your account should include a discussion of natural statistics and their properties and of various conditional tests on natural parameters.

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• A1.11 B1.16

  Stochastic Financial Models

  Part II, 2003

  (i) In the context of a single-period financial market with $d$ traded assets, what is an arbitrage? What is an equivalent martingale measure?

  A simple single-period financial market contains two assets, $S^{0}$ (a bond), and $S^{1}$ (a share). The period can be good, bad, or indifferent, with probabilities $1 / 3$ each. At the beginning of the period, time 0 , both assets are worth 1 , i.e.

  $$S_{0}^{0}=1=S_{0}^{1}$$

  and at the end of the period, time 1 , the share is worth

  $$S_{1}^{1}= \begin{cases}a & \text { if the period was bad, } \\ b & \text { if the period was indifferent } \\ c & \text { if the period was good }\end{cases}$$

  where $a<b<c$. The bond is always worth 1 at the end of the period. Show that there is no arbitrage in this market if and only if $a<1<c$.

  (ii) An agent with $C^{2}$ strictly increasing strictly concave utility $U$ has wealth $w_{0}$ at time 0 , and wishes to invest his wealth in shares and bonds so as to maximise his expected utility of wealth at time 1 . Explain how the solution to his optimisation problem generates an equivalent martingale measure.

  Assume now that $a=3 / 4, b=1$, and $c=3 / 2$. Characterise all equivalent martingale measures for this problem. Characterise all equivalent martingale measures which arise as solutions of an agent's optimisation problem.

  Calculate the largest and smallest possible prices for a European call option with strike 1 and expiry 1, as the pricing measure ranges over all equivalent martingale measures. Calculate the corresponding bounds when the pricing measure is restricted to the set arising from expected-utility-maximising agents' optimisation problems.

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• A3.11 B3.16

  Stochastic Financial Models

  Part II, 2003

  (i) What does it mean to say that the process $\left(W_{t}\right)_{t \geqslant 0}$ is a Brownian motion? What does it mean to say that the process $\left(M_{t}\right)_{t \geqslant 0}$ is a martingale?

  Suppose that $\left(W_{t}\right)_{t \geqslant 0}$ is a Brownian motion and the process $\left(X_{t}\right)_{t \geqslant 0}$ is given in terms of $W$ as

  $$X_{t}=x_{0}+\sigma W_{t}+\mu t$$

  for constants $\sigma, \mu$. For what values of $\theta$ is the process

  $$M_{t}=\exp \left(\theta X_{t}-\lambda t\right)$$

  a martingale? (Here, $\lambda$ is a positive constant.)

  (ii) In a standard Black-Scholes model, the price at time $t$ of a share is represented as $S_{t}=\exp \left(X_{t}\right)$. You hold a perpetual American put option on this share, with strike $K$; you may exercise at any stopping time $\tau$, and upon exercise you receive $\max \left\{0, K-S_{\tau}\right\}$. Let $0<a<\log K$. Suppose you plan to use the exercise policy: 'Exercise as soon as the price falls to $e^{a}$ or lower.' Calculate what the option would be worth if you were to follow this policy. (Assume that the riskless rate of interest is constant and equal to $r>0$.) For what choice of $a$ is this value maximised?

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• A4.12 B4.16

  Stochastic Financial Models

  Part II, 2003

  A single-period market contains $d$ risky assets, $S^{1}, S^{2}, \ldots, S^{d}$, initially worth $\left(S_{0}^{1}, S_{0}^{2}, \ldots, S_{0}^{d}\right)$, and at time 1 worth random amounts $\left(S_{1}^{1}, S_{1}^{2}, \ldots, S_{1}^{d}\right)$ whose first two moments are given by

  $$\mu=E S_{1}, \quad V=\operatorname{cov}\left(S_{1}\right) \equiv E\left[\left(S_{1}-E S_{1}\right)\left(S_{1}-E S_{1}\right)^{T}\right]$$

  An agent with given initial wealth $w_{0}$ is considering how to invest in the available assets, and has asked for your advice. Develop the theory of the mean-variance efficient frontier far enough to exhibit explicitly the minimum-variance portfolio achieving a required mean return, assuming that $V$ is non-singular. How does your analysis change if a riskless asset $S^{0}$ is added to the market? Under what (sufficient) conditions would an agent maximising expected utility actually choose a portfolio on the mean-variance efficient frontier?

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• A2.13 B2.21

  Foundations of Quantum Mechanics

  Part II, 2003

  (i) Define the Heisenberg picture of quantum mechanics in relation to the Schrödinger picture. Explain how the two pictures provide equivalent descriptions of observable results.

  Derive the equation of motion for an operator in the Heisenberg picture.

  (ii) For a particle moving in one dimension, the Hamiltonian is

  $$\hat{H}=\frac{\hat{p}^{2}}{2 m}+V(\hat{x}),$$

  where $\hat{x}$ and $\hat{p}$ are the position and momentum operators, and the state vector is $|\Psi\rangle$.

  Eigenstates of $\hat{x}$ and $\hat{p}$ satisfy

  $$\langle x \mid p\rangle=\left(\frac{1}{2 \pi \hbar}\right)^{1 / 2} e^{i p x / \hbar}, \quad\left\langle x \mid x^{\prime}\right\rangle=\delta\left(x-x^{\prime}\right), \quad\left\langle p \mid p^{\prime}\right\rangle=\delta\left(p-p^{\prime}\right) .$$

  Use standard methods in the Dirac formalism to show that

  $$\begin{aligned} &\left\langle x|\hat{p}| x^{\prime}\right\rangle=-i \hbar \frac{\partial}{\partial x} \delta\left(x-x^{\prime}\right) \\ &\left\langle p|\hat{x}| p^{\prime}\right\rangle=i \hbar \frac{\partial}{\partial p} \delta\left(p-p^{\prime}\right) \end{aligned}$$

  Calculate $\left\langle x|\hat{H}| x^{\prime}\right\rangle$ and express $\langle x|\hat{p}| \Psi\rangle,\langle x|\hat{H}| \Psi\rangle$ in terms of the position space wave function $\Psi(x)$.

  Compute the momentum space Hamiltonian for the harmonic oscillator with potential $V(\hat{x})=\frac{1}{2} m \omega^{2} \hat{x}^{2}$.

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• A3.13 B3.21

  Foundations of Quantum Mechanics

  Part II, 2003

  (i) What are the commutation relations satisfied by the components of an angular momentum vector $\mathbf{J}$ ? State the possible eigenvalues of the component $J_{3}$ when $\mathbf{J}^{2}$ has eigenvalue $j(j+1) \hbar^{2}$.

  Describe how the Pauli matrices

  $$\sigma_{1}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_{2}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_{3}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)$$

  are used to construct the components of the angular momentum vector $\mathbf{S}$ for a spin $\frac{1}{2}$ system. Show that they obey the required commutation relations.

  Show that $S_{1}, S_{2}$ and $S_{3}$ each have eigenvalues $\pm \frac{1}{2} \hbar$. Verify that $\mathbf{S}^{2}$ has eigenvalue $\frac{3}{4} \hbar^{2} .$

  (ii) Let $\mathbf{J}$ and $|j m\rangle$ denote the standard operators and state vectors of angular momentum theory. Assume units where $\hbar=1$. Consider the operator

  $$U(\theta)=e^{-i \theta J_{2}}$$

  Show that

  $$\begin{aligned} &U(\theta) J_{1} U(\theta)^{-1}=\cos \theta J_{1}-\sin \theta J_{3} \\ &U(\theta) J_{3} U(\theta)^{-1}=\sin \theta J_{1}+\cos \theta J_{3} \end{aligned}$$

  Show that the state vectors $U\left(\frac{\pi}{2}\right)|j m\rangle$ are eigenvectors of $J_{1}$. Suppose that $J_{1}$ is measured for a system in the state $|j m\rangle$; show that the probability that the result is $m^{\prime}$ equals

  $$\left|\left\langle j m^{\prime}\left|e^{i \frac{\pi}{2} J_{2}}\right| j m\right\rangle\right|^{2}$$

  Consider the case $j=m=\frac{1}{2}$. Evaluate the probability that the measurement of $J_{1}$ will result in $m^{\prime}=-\frac{1}{2}$.

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• A4.15 B4.22

  Foundations of Quantum Mechanics

  Part II, 2003

  Discuss the quantum mechanics of the one-dimensional harmonic oscillator using creation and annihilation operators, showing how the energy levels are calculated.

  A quantum mechanical system consists of two interacting harmonic oscillators and has the Hamiltonian

  $$H=\frac{1}{2} \hat{p}_{1}^{2}+\frac{1}{2} \hat{x}_{1}^{2}+\frac{1}{2} \hat{p}_{2}^{2}+\frac{1}{2} \hat{x}_{2}^{2}+\lambda \hat{x}_{1} \hat{x}_{2}$$

  For $\lambda=0$, what are the degeneracies of the three lowest energy levels? For $\lambda \neq 0$ compute, to lowest non-trivial order in perturbation theory, the energies of the ground state and first excited state.

  [Standard results for perturbation theory may be stated without proof.]

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• A1.15 B1.24

  General Relativity

  Part II, 2003

  (i) The worldline $x^{a}(\lambda)$ of a massive particle moving in a spacetime with metric $g_{a b}$ obeys the geodesic equation

  $$\frac{d^{2} x^{a}}{d \tau^{2}}+\Gamma^a_{bc} \frac{d x^{b}}{d \tau} \frac{d x^{c}}{d \tau}=0$$

  where $\tau$ is the particle's proper time and $\Gamma^a_{bc}$ are the Christoffel symbols; these are the equations of motion for the Lagrangian

  $$L_{1}=-m \sqrt{-g_{a b} \dot{x}^{a} \dot{x}^{b}}$$

  where $m$ is the particle's mass, and $\dot{x}^{a}=d x^{a} / d \lambda$. Why is the choice of worldline parameter $\lambda$ irrelevant? Among all possible worldlines passing through points $A$ and $B$, why is $x^{a}(\lambda)$ the one that extremizes the proper time elapsed between $A$ and $B$ ?

  Explain how the equations of motion for a massive particle may be obtained from the alternative Lagrangian

  $$L_{2}=\frac{1}{2} g_{a b} \dot{x}^{a} \dot{x}^{b} .$$

  What can you conclude from the fact that $L_{2}$ has no explicit dependence on $\lambda$ ? How are the equations of motion for a massless particle obtained from $L_{2}$ ?

  (ii) A photon moves in the Schwarzschild metric

  $$d s^{2}=\left(1-\frac{2 M}{r}\right)^{-1} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)-\left(1-\frac{2 M}{r}\right) d t^{2}$$

  Given that the motion is confined to the plane $\theta=\pi / 2$, obtain the radial equation

  $$\left(\frac{d r}{d \lambda}\right)^{2}=E^{2}-\frac{h^{2}}{r^{2}}\left(1-\frac{2 M}{r}\right)$$

  where $E$ and $h$ are constants, the physical meaning of which should be stated.

  Setting $u=1 / r$, obtain the equation

  $$\frac{d^{2} u}{d \phi^{2}}+u=3 M u^{2}$$

  Using the approximate solution

  $$u=\frac{1}{b} \sin \phi+\frac{M}{2 b^{2}}(3+\cos 2 \phi)+\ldots$$

  obtain the standard formula for the deflection of light passing far from a body of mass $M$ with impact parameter $b$. Reinstate factors of $G$ and $c$ to give your result in physical units.

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• A2.15 B2.23

  General Relativity

  Part II, 2003

  (i) What is a "stationary" metric? What distinguishes a stationary metric from a "static" metric?

  A Killing vector field $K^{a}$ of a metric $g_{a b}$ satisfies

  $$K_{a ; b}+K_{b ; a}=0$$

  Show that this is equivalent to

  $$g_{a b, c} K^{c}+g_{a c} K_{, b}^{c}+g_{c b} K_{, a}^{c}=0$$

  Hence show that a constant vector field $K^{a}$ with one non-zero component, $K^{4}$ say, is a Killing vector field if $g_{a b}$ is independent of $x^{4}$.

  (ii) Given that $K^{a}$ is a Killing vector field, show that $K_{a} u^{a}$ is constant along the geodesic worldline of a massive particle with 4-velocity $u^{a}$. Hence find the energy $\varepsilon$ of a particle of unit mass moving in a static spacetime with metric

  $$d s^{2}=h_{i j} d x^{i} d x^{j}-e^{2 U} d t^{2}$$

  where $h_{i j}$ and $U$ are functions only of the space coordinates $x^{i}$. By considering a particle with speed small compared with that of light, and given that $U \ll 1$, show that $h_{i j}=\delta_{i j}$ to lowest order in the Newtonian approximation, and that $U$ is the Newtonian potential.

  A metric admits an antisymmetric tensor $Y_{a b}$ satisfying

  $$Y_{a b ; c}+Y_{a c ; b}=0$$

  Given a geodesic $x^{a}(\lambda)$, let $s_{a}=Y_{a b} \dot{x}^{b}$. Show that $s_{a}$ is parallelly propagated along the geodesic, and that it is orthogonal to the tangent vector of the geodesic. Hence show that the scalar

  $$\phi=s^{a} s_{a}$$

  is constant along the geodesic.

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• A4.17 B4.25

  General Relativity

  Part II, 2003

  What are "inertial coordinates" and what is their physical significance? [A proof of the existence of inertial coordinates is not required.] Let $O$ be the origin of inertial coordinates and let $\left.R_{a b c d}\right|_{O}$ be the curvature tensor at $O$ (with all indices lowered). Show that $\left.R_{a b c d}\right|_{O}$ can be expressed entirely in terms of second partial derivatives of the metric $g_{a b}$, evaluated at $O$. Use this expression to deduce that (a) $R_{a b c d}=-R_{b a c d}$ (b) $R_{a b c d}=R_{c d a b}$ (c) $R_{a[b c d]}=0$.

  Starting from the expression for $R_{b c d}^{a}$ in terms of the Christoffel symbols, show (again by using inertial coordinates) that

  $$R_{a b[c d ; e]}=0$$

  Obtain the contracted Bianchi identities and explain why the Einstein equations take the form

  $$R_{a b}-\frac{1}{2} R g_{a b}=8 \pi T_{a b}-\Lambda g_{a b},$$

  where $T_{a b}$ is the energy-momentum tensor of the matter and $\Lambda$ is an arbitrary constant.

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• A1.20 B1.20

  Numerical Analysis

  Part II, 2003

  (i) The linear algebraic equations $A \mathbf{u}=\mathbf{b}$, where $A$ is symmetric and positive-definite, are solved with the Gauss-Seidel method. Prove that the iteration always converges.

  (ii) The Poisson equation $\nabla^{2} u=f$ is given in the bounded, simply connected domain $\Omega \subseteq \mathbb{R}^{2}$, with zero Dirichlet boundary conditions on $\partial \Omega$. It is approximated by the fivepoint formula

  $$U_{m-1, n}+U_{m, n-1}+U_{m+1, n}+U_{m, n+1}-4 U_{m, n}=(\Delta x)^{2} f_{m, n}$$

  where $U_{m, n} \approx u(m \Delta x, n \Delta x), \quad f_{m, n}=f(m \Delta x, n \Delta x)$, and $(m \Delta x, n \Delta x)$ is in the interior of $\Omega$.

  Assume for the sake of simplicity that the intersection of $\partial \Omega$ with the grid consists only of grid points, so that no special arrangements are required near the boundary. Prove that the method can be written in a vector notation, $A \mathbf{u}=\mathbf{b}$ with a negative-definite matrix $A$.

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• A2.19 B2.19

  Numerical Analysis

  Part II, 2003

  (i) Explain briefly what is meant by the convergence of a numerical method for ordinary differential equations.

  (ii) Suppose the sufficiently-smooth function $f: \mathbb{R} \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ obeys the Lipschitz condition: there exists $\lambda>0$ such that

  $$\|\mathbf{f}(t, \mathbf{x})-\mathbf{f}(t, \mathbf{y})\| \leqslant \lambda\|\mathbf{x}-\mathbf{y}\|, \quad \mathbf{x}, \mathbf{y} \in \mathbb{R}^{d}, t \geqslant 0$$

  Prove from first principles, without using the Dahlquist equivalence theorem, that the trapezoidal rule

  $$\mathbf{y}_{n+1}=\mathbf{y}_{n}+\frac{1}{2} h\left[\mathbf{f}\left(t_{n}, \mathbf{y}_{n}\right)+\mathbf{f}\left(t_{n+1}, \mathbf{y}_{n+1}\right)\right]$$

  for the solution of the ordinary differential equation

  $$\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y}), \quad t \geqslant 0, \quad \mathbf{y}(0)=\mathbf{y}_{0},$$

  converges.

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• A3.19 B3.20

  Numerical Analysis

  Part II, 2003

  (i) The diffusion equation

  $$\frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\left(a(x) \frac{\partial u}{\partial x}\right), \quad 0 \leqslant x \leqslant 1, \quad t \geqslant 0$$

  with the initial condition $u(x, 0)=\phi(x), 0 \leqslant x \leqslant 1$ and zero boundary conditions at $x=0$ and $x=1$, is solved by the finite-difference method

  $$\begin{gathered} u_{m}^{n+1}=u_{m}^{n}+\mu\left[a_{m-\frac{1}{2}} u_{m-1}^{n}-\left(a_{m-\frac{1}{2}}+a_{m+\frac{1}{2}}\right) u_{m}^{n}+a_{m+\frac{1}{2}} u_{m+1}^{n}\right] \\ m=1,2, \ldots, N \end{gathered}$$

  where $\mu=\Delta t /(\Delta x)^{2}, \quad \Delta x=\frac{1}{N+1}$ and $u_{m}^{n} \approx u(m \Delta x, n \Delta t), a_{\alpha}=a(\alpha \Delta x)$.

  Assuming sufficient smoothness of the function $a$, and that $\mu$ remains constant as $\Delta x>0$ and $\Delta t>0$ become small, prove that the exact solution satisfies the numerical scheme with error $O\left((\Delta x)^{3}\right)$.

  (ii) For the problem defined in Part (i), assume that there exist $0<a_{-}<a_{+}<\infty$ such that $a_{-} \leqslant a(x) \leqslant a_{+}, \quad 0 \leqslant x \leqslant 1$. Prove that the method is stable for $0<\mu \leqslant 1 /\left(2 a_{+}\right)$.

  [Hint: You may use without proof the Gerschgorin theorem: All the eigenvalues of the matrix $A=\left(a_{k, l}\right)_{k, l=1, \ldots, M}$ are contained in $\bigcup_{k=1}^{m} \mathbb{S}_{k}$, where

  $$\left.\mathbb{S}_{k}=\left\{z \in \mathbb{C}:\left|z-a_{k, k}\right| \leqslant \sum_{\substack{l=1 \\ l \neq k}}^{m}\left|a_{k, l}\right|\right\}, \quad k=1,2, \ldots, m . \quad\right]$$

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• A4.23 B4.20

  Numerical Analysis

  Part II, 2003

  Write an essay on the conjugate gradient method. Your essay should include:

  (a) a statement of the method and a sketch of its derivation;

  (b) discussion, without detailed proofs, but with precise statements of relevant theorems, of the conjugacy of the search directions;

  (c) a description of the standard form of the algorithm;

  (d) discussion of the connection of the method with Krylov subspaces.

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• B1.5

  Combinatorics

  Part II, 2003

  Let $G$ be a graph of order $n \geqslant 4$. Prove that if $G$ has $t_{2}(n)+1$ edges then it contains two triangles with a common edge. Here, $t_{2}(n)=\left\lfloor n^{2} / 4\right\rfloor$ is the Turán number.

  Suppose instead that $G$ has exactly one triangle. Show that $G$ has at most $t_{2}(n-1)+2$ edges, and that this number can be attained.

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• B2.5

  Combinatorics

  Part II, 2003

  Prove Ramsey's theorem in its usual infinite form, namely, that if $\mathbb{N}^{(r)}$ is finitely coloured then there is an infinite subset $M \subset \mathbb{N}$ such that $M^{(r)}$ is monochromatic.

  Now let the graph $\mathbb{N}^{(2)}$ be coloured with an infinite number of colours in such a way that there is no infinite $M \subset \mathbb{N}$ with $M^{(2)}$ monochromatic. By considering a suitable 2-colouring of the set $\mathbb{N}^{(4)}$ of 4 -sets, show that there is an infinite $M \subset \mathbb{N}$ with the property that any two edges of $M^{(2)}$ of the form $a d, b c$ with $a<b<c<d$ have different colours.

  By considering two further 2-colourings of $\mathbb{N}^{(4)}$, show that there is an infinite $M \subset \mathbb{N}$ such that any two non-incident edges of $M^{(2)}$ have different colours.

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• B4.1

  Combinatorics

  Part II, 2003

  Write an essay on the Kruskal-Katona theorem. As well as stating the theorem and giving a detailed sketch of a proof, you should describe some further results that may be derived from it.

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• B1.6

  Representation Theory

  Part II, 2003

  Define the inner product $\langle\varphi, \psi\rangle$ of two class functions from the finite group $G$ into the complex numbers. Prove that characters of the irreducible representations of $G$ form an orthonormal basis for the space of class functions.

  Consider the representation $\pi: S_{n} \rightarrow G L_{n}(\mathbb{C})$ of the symmetric group $S_{n}$ by permutation matrices. Show that $\pi$ splits as a direct sum $1 \oplus \rho$ where 1 denotes the trivial representation. Is the $(n-1)$-dimensional representation $\rho$ irreducible?

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• B2.6

  Representation Theory

  Part II, 2003

  Let $V_{n}$ be the space of homogeneous polynomials of degree $n$ in two variables $z_{1}$ and $z_{2}$. Define a left action of $G=S U_{2}$ on the space of polynomials by setting

  $$(g P) z=P(z g),$$

  where $P \in \mathbb{C}\left[z_{1}, z_{2}\right], \quad g=\left(\begin{array}{cc}a & b \\ c & d\end{array}\right), \quad z=\left(z_{1}, z_{2}\right) \quad$ and $\quad z g=\left(a z_{1}+c z_{2}, b z_{1}+d z_{2}\right)$.

  Show that

  (a) the representations $V_{n}$ are irreducible,

  (b) the representations $V_{n}$ exhaust the irreducible representations of $S U_{2}$, and

  

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• B3.5

  Representation Theory

  Part II, 2003

  If $\rho_{1}: G_{1} \rightarrow G L\left(V_{1}\right)$ and $\rho_{2}: G_{2} \rightarrow G L\left(V_{2}\right)$ are representations of the finite groups $G_{1}$ and $G_{2}$ respectively, define the tensor product $\rho_{1} \otimes \rho_{2}$ as a representation of the group $G_{1} \times G_{2}$ and show that its character is given by

  $$\chi_{\rho_{1} \otimes \rho_{2}}\left(g_{1}, g_{2}\right)=\chi_{\rho_{1}}\left(g_{1}\right) \chi_{\rho_{2}}\left(g_{2}\right)$$

  Prove that

  (a) if $\rho_{1}$ and $\rho_{2}$ are irreducible, then $\rho_{1} \otimes \rho_{2}$ is an irreducible representation of $G_{1} \times G_{2}$;

  (b) each irreducible representation of $G_{1} \times G_{2}$ is equivalent to a representation $\rho_{1} \otimes \rho_{2}$ where each $\rho_{i}$ is irreducible $(i=1,2) .$

  Is every representation of $G_{1} \times G_{2}$ the tensor product of a representation of $G_{1}$ and a representation of $G_{2}$ ?

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• B4.2

  Representation Theory

  Part II, 2003

  Assume that the group $S L_{2}\left(\mathbb{F}_{3}\right)$ of $2 \times 2$ matrices of determinant 1 with entries from the field $\mathbb{F}_{3}$ has presentation

  $$\left\langle X, P, Q: X^{3}=P^{4}=1, P^{2}=Q^{2}, P Q P^{-1}=Q^{-1}, X P X^{-1}=Q, X Q X^{-1}=P Q\right\rangle$$

  Show that the subgroup generated by $P^{2}$ is central and that the quotient group can be identified with the alternating group $A_{4}$. Assuming further that $S L_{2}\left(\mathbb{F}_{3}\right)$ has seven conjugacy classes find the character table.

  Is it true that every irreducible character is induced up from the character of a 1-dimensional representation of some subgroup?

  [Hint: You may find it useful to note that $S L_{2}\left(\mathbb{F}_{3}\right)$ may be regarded as a subgroup of $S U_{2}$, providing a faithful 2-dimensional representation; the subgroup generated by $P$ and $Q$ is the quaternion group of order 8 , acting irreducibly.]

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• B1.7

  Galois Theory

  Part II, 2003

  What does it mean to say that a field is algebraically closed? Show that a field $M$ is algebraically closed if and only if, for any finite extension $L / K$ and every homomorphism $\sigma: K \hookrightarrow M$, there exists a homomorphism $L \hookrightarrow M$ whose restriction to $K$ is $\sigma$.

  Let $K$ be a field of characteristic zero, and $M / K$ an algebraic extension such that every nonconstant polynomial over $K$ has at least one root in $M$. Prove that $M$ is algebraically closed.

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• B3.6

  Galois Theory

  Part II, 2003

  Let $f$ be a separable polynomial of degree $n \geqslant 1$ over a field $K$. Explain what is meant by the Galois group $\operatorname{Gal}(f / K)$ of $f$ over $K$. Explain how $\operatorname{Gal}(f / K)$ can be identified with a subgroup of the symmetric group $S_{n}$. Show that as a permutation group, $\operatorname{Gal}(f / K)$ is transitive if and only if $f$ is irreducible over $K$.

  Show that the Galois group of $f(X)=X^{5}+20 X^{2}-2$ over $\mathbb{Q}$ is $S_{5}$, stating clearly any general results you use.

  Now let $K / \mathbb{Q}$ be a finite extension of prime degree $p>5$. By considering the degrees of the splitting fields of $f$ over $K$ and $\mathbb{Q}$, show that $\operatorname{Gal}(f / K)=S_{5}$ also.

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• B4.3

  Galois Theory

  Part II, 2003

  Write an essay on finite fields and their Galois theory.

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• B1.8

  Differentiable Manifolds

  Part II, 2003

  State the Implicit Function Theorem and outline how it produces submanifolds of Euclidean spaces.

  Show that the unitary group $U(n) \subset G L(n, \mathbb{C})$ is a smooth manifold and find its dimension.

  Identify the tangent space to $U(n)$ at the identity matrix as a subspace of the space of $n \times n$ complex matrices.

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• B2.7

  Differentiable Manifolds

  Part II, 2003

  Let $M$ and $N$ be smooth manifolds. If $\pi: M \times N \rightarrow M$ is the projection onto the first factor and $\pi^{*}$ is the map in cohomology induced by the pull-back map on differential forms, show that $\pi^{*}\left(H^{k}(M)\right)$ is a direct summand of $H^{k}(M \times N)$ for each $k \geqslant 0$.

  Taking $H^{k}(M)$ to be zero for $k<0$ and $k>\operatorname{dim} M$, show that for $n \geqslant 1$ and all $k$

  $$H^{k}\left(M \times S^{n}\right) \cong H^{k}(M) \oplus H^{k-n}(M)$$

  [You might like to use induction in n.]

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• B4.4

  Differentiable Manifolds

  Part II, 2003

  Define the 'pull-back' homomorphism of differential forms determined by the smooth map $f: M \rightarrow N$ and state its main properties.

  If $\theta: W \rightarrow V$ is a diffeomorphism between open subsets of $\mathbb{R}^{m}$ with coordinates $x_{i}$ on $V$ and $y_{j}$ on $W$ and the $m$-form $\omega$ is equal to $f d x_{1} \wedge \ldots \wedge d x_{m}$ on $V$, state and prove the expression for $\theta^{*}(\omega)$ as a multiple of $d y_{1} \wedge \ldots \wedge d y_{m}$.

  Define the integral of an $m$-form $\omega$ over an oriented $m$-manifold $M$ and prove that it is well-defined.

  Show that the inclusion map $f: N \hookrightarrow M$, of an oriented $n$-submanifold $N$ (without boundary) into $M$, determines an element $\nu$ of $H_{n}(M) \cong \operatorname{Hom}\left(H^{n}(M), \mathbb{R}\right)$. If $M=N \times P$ and $f(x)=(x, p)$, for $x \in N$ and $p$ fixed in $P$, what is the relation between $\nu$ and $\pi^{*}\left(\left[\omega_{N}\right]\right)$, where $\left[\omega_{N}\right]$ is the fundamental cohomology class of $N$ and $\pi$ is the projection onto the first factor?

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• B2.8

  Algebraic Topology

  Part II, 2003

  Define the fundamental group of a topological space and explain briefly why a continuous map gives rise to a homomorphism between fundamental groups.

  Let $X$ be a subspace of the Euclidean space $\mathbb{R}^{3}$ which contains all of the points $(x, y, 0)$ with $(x, y) \neq(0,0)$, and which does not contain any of the points $(0,0, z)$. Show that $X$ has an infinite fundamental group.

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• B3.7

  Algebraic Topology

  Part II, 2003

  Define a covering map. Prove that any covering map induces an injective homomorphisms of fundamental groups.

  Show that there is a non-trivial covering map of the real projective plane. Explain how to use this to find the fundamental group of the real projective plane.

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• B2.9

  Number Fields

  Part II, 2003

  By Dedekind's theorem, or otherwise, factorise $2,3,5$ and 7 into prime ideals in the field $K=\mathbb{Q}(\sqrt{-34})$. Show that the ideal equations

  $$[\omega]=[5, \omega][7, \omega], \quad[\omega+3]=[2, \omega+3][5, \omega+3]^{2}$$

  hold in $K$, where $\omega=1+\sqrt{-34}$. Hence, prove that the ideal class group of $K$ is cyclic of order $4 .$

  [It may be assumed that the Minkowski constant for $K$ is $2 / \pi$.]

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• B4.5

  Algebraic Topology

  Part II, 2003

  State the Mayer-Vietoris theorem. You should give the definition of all the homomorphisms involved.

  Compute the homology groups of the union of the 2 -sphere with the line segment from the North pole to the South pole.

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• B1.9

  Number Fields

  Part II, 2003

  Let $K=\mathbb{Q}(\alpha)$, where $\alpha=\sqrt[3]{10}$, and let $\mathcal{O}_{K}$ be the ring of algebraic integers of $K$. Show that the field polynomial of $r+s \alpha$, with $r$ and $s$ rational, is $(x-r)^{3}-10 s^{3}$.

  Let $\beta=\frac{1}{3}\left(\alpha^{2}+\alpha+1\right)$. By verifying that $\beta=3 /(\alpha-1)$ and determining the field polynomial, or otherwise, show that $\beta$ is in $\mathcal{O}_{K}$.

  By computing the traces of $\theta, \alpha \theta, \alpha^{2} \theta$, show that the elements of $\mathcal{O}_{K}$ have the form

  $$\theta=\frac{1}{3}\left(l+\frac{1}{10} m \alpha+\frac{1}{10} n \alpha^{2}\right)$$

  where $l, m, n$ are integers. By further computing the norm of $\frac{1}{10} \alpha(m+n \alpha)$, show that $\theta$ can be expressed as $\frac{1}{3}(u+v \alpha)+w \beta$ with $u, v, w$ integers. Deduce that $1, \alpha, \beta$ form an integral basis for $K$.

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• B4.6

  Number Fields

  Part II, 2003

  Write an essay on the Dirichlet unit theorem with particular reference to quadratic fields.

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• B1.10

  Hilbert Spaces

  Part II, 2003

  Let $H$ be a Hilbert space and let $T \in \mathcal{B}(H)$.

  (a) Define what it means for $T$ to be (i) invertible, and (ii) bounded below. Prove that $T$ is invertible if and only if both $T$ and $T^{*}$ are bounded below.

  (b) Define what it means for $T$ to be normal. Prove that $T$ is normal if and only if $\|T x\|=\left\|T^{*} x\right\|$ for all $x \in H$. Deduce that, if $T$ is normal, then every point of Sp $T$ is an approximate eigenvalue of $T$.

  (c) Let $S \in \mathcal{B}(H)$ be a self-adjoint operator, and let $\left(x_{n}\right)$ be a sequence in $H$ such that $\left\|x_{n}\right\|=1$ for all $n$ and $\left\|S x_{n}\right\| \rightarrow\|S\|$ as $n \rightarrow \infty$. Show, by direct calculation, that

  $$\left\|\left(S^{2}-\|S\|^{2}\right) x_{n}\right\|^{2} \rightarrow 0 \quad \text { as } n \rightarrow \infty$$

  and deduce that at least one of $\pm\|S\|$ is an approximate eigenvalue of $S$.

  (d) Deduce that, with $S$ as in (c),

  $$r(S)=\|S\|=\sup \{|\langle S x, x\rangle|: x \in H,\|x\|=1\}$$

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• B3.8

  Hilbert Spaces

  Part II, 2003

  Let $\mathcal{H}$ be the space of all functions on the real line $\mathbb{R}$ of the form $p(x) e^{-x^{2} / 2}$, where $p$ is a polynomial with complex coefficients. Make $\mathcal{H}$ into an inner-product space, in the usual way, by defining the inner product to be

  $$\langle f, g\rangle=\int_{-\infty}^{\infty} f(t) \overline{g(t)} d t, \quad f, g \in \mathcal{H}$$

  You should assume, without proof, that this equation does define an inner product on $\mathcal{H}$. Define the norm by $\|f\|_{2}=\langle f, f\rangle^{1 / 2}$ for $f \in \mathcal{H}$. Now define a sequence of functions $\left(F_{n}\right)_{n \geqslant 0}$ on $\mathbb{R}$ by

  $$F_{n}(x)=(-1)^{n} e^{x^{2} / 2} \frac{d^{n}}{d x^{n}} e^{-x^{2}}$$

  Prove that $\left(F_{n}\right)$ is an orthogonal sequence in $\mathcal{H}$ and that it spans $\mathcal{H}$.

  For every $f \in \mathcal{H}$ define the Fourier transform $\widehat{f}$ of $f$ by

  $$\widehat{f}(t)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(x) e^{-i t x} d x, \quad t \in \mathbb{R}$$

  Show that

  (a) $\widehat{F}_{n}=(-i)^{n} F_{n}$ for $n=0,1,2, \ldots$;

  (b) for all $f \in \mathcal{H}$ and $x \in \mathbb{R}$,

  $$\widehat{\widehat{f}}(x)=f(-x)$$

  (c) $\|\widehat{f}\|_{2}=\|f\|_{2}$ for all $f \in \mathcal{H}$.

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• B4.7

  Hilbert Spaces

  Part II, 2003

  Let $H$ be a Hilbert space and let $T \in \mathcal{B}(H)$.

  (a) Show that if $\|I-T\|<1$ then $T$ is invertible.

  (b) Prove that if $T$ is invertible and if $S \in \mathcal{B}(H)$ satisfies $\|S-T\|<\left\|T^{-1}\right\|^{-1}$, then $S$ is invertible.

  (c) Define what it means for $T$ to be compact. Prove that the set of compact operators on $H$ is a closed subset of $\mathcal{B}(H)$.

  (d) Prove that $T$ is compact if and only if there is a sequence $\left(F_{n}\right)$ in $\mathcal{B}(H)$, where each operator $F_{n}$ has finite rank, such that $\left\|F_{n}-T\right\| \rightarrow 0$ as $n \rightarrow \infty$.

  (e) Suppose that $T=A+K$, where $A$ is invertible and $K$ is compact. Prove that then, also, $T=B+F$, where $B$ is invertible and $F$ has finite rank.

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• B1.11

  Riemann Surfaces

  Part II, 2003

  Prove that a holomorphic map from $\mathbb{P}^{1}$ to itself is either constant or a rational function. Prove that a holomorphic map of degree 1 from $\mathbb{P}^{1}$ to itself is a Möbius transformation.

  Show that, for every finite set of distinct points $z_{1}, z_{2}, \ldots, z_{N}$ in $\mathbb{P}^{1}$ and any values $w_{1}, w_{2}, \ldots, w_{N} \in \mathbb{P}^{1}$, there is a holomorphic function $f: \mathbb{P}^{1} \rightarrow \mathbb{P}^{1}$ with $f\left(z_{n}\right)=w_{n}$ for $n=1,2, \ldots, N$.

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• B3.9

  Riemann Surfaces

  Part II, 2003

  Let $L$ be the lattice $\mathbb{Z} \omega_{1}+\mathbb{Z} \omega_{2}$ for two non-zero complex numbers $\omega_{1}, \omega_{2}$ whose ratio is not real. Recall that the Weierstrass function $\wp$ is given by the series

  $$\wp(u)=\frac{1}{u^{2}}+\sum_{\omega \in L-\{0\}}\left(\frac{1}{(u-\omega)^{2}}-\frac{1}{\omega^{2}}\right)$$

  the function $\zeta$ is the (unique) odd anti-derivative of $-\wp$; and $\sigma$ is defined by the conditions

  $$\sigma^{\prime}(u)=\zeta(u) \sigma(u) \quad \text { and } \quad \sigma^{\prime}(0)=1$$

  (a) By writing a differential equation for $\sigma(-u)$, or otherwise, show that $\sigma$ is an odd function.

  (b) Show that $\sigma\left(u+\omega_{i}\right)=-\sigma(u) \exp \left(a_{i}\left(u+b_{i}\right)\right)$ for some constants $a_{i}, b_{i}$. Use (a) to express $b_{i}$ in terms of $\omega_{i}$. [Do not attempt to express $a_{i}$ in terms of $\omega_{i}$.]

  (c) Show that the function $f(u)=\sigma(2 u) / \sigma(u)^{4}$ is periodic with respect to the lattice $L$ and deduce that $f(u)=-\wp^{\prime}(u)$.

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• B4.8

  Riemann Surfaces

  Part II, 2003

  (a) Define the degree $\operatorname{deg} f$ of a meromorphic function on the Riemann sphere $\mathbb{P}^{1}$. State the Riemann-Hurwitz theorem.

  Let $f$ and $g$ be two rational functions on the sphere $\mathbb{P}^{1}$. Show that

  $$\operatorname{deg}(f+g) \leqslant \operatorname{deg} f+\operatorname{deg} g$$

  Deduce that

  $$|\operatorname{deg} f-\operatorname{deg} g| \leqslant \operatorname{deg}(f+g) \leqslant \operatorname{deg} f+\operatorname{deg} g .$$

  (b) Describe the topological type of the Riemann surface defined by the equation $w^{2}+2 w=z^{5}$ in $\mathbb{C}^{2}$. [You should analyse carefully the behaviour as $w$ and $z$ approach $\infty$.]

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• B2.10

  Algebraic Curves

  Part II, 2003

  (a) For which polynomials $f(x)$ of degree $d>0$ does the equation $y^{2}=f(x)$ define a smooth affine curve?

  (b) Now let $C$ be the completion of the curve defined in (a) to a projective curve. For which polynomials $f(x)$ of degree $d>0$ is $C$ a smooth projective curve?

  (c) Suppose that $C$, defined in (b), is a smooth projective curve. Consider a map $p: C \rightarrow \mathbb{P}^{1}$, given by $p(x, y)=x$. Find the degree and the ramification points of $p$.

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• B3.10

  Algebraic Curves

  Part II, 2003

  (a) Let $X \subseteq \mathbb{A}^{n}$ be an affine algebraic variety. Define the tangent space $T_{p} X$ for $p \in X$. Show that the set

  $$\left\{p \in X \mid \operatorname{dim} T_{p} X \geqslant d\right\}$$

  is closed, for every $d \geqslant 0$.

  (b) Let $C$ be an irreducible projective curve, $p \in C$, and $f: C \backslash\{p\} \rightarrow \mathbb{P}^{n}$ a rational map. Show, carefully quoting any theorems that you use, that if $C$ is smooth at $p$ then $f$ extends to a regular map at $p$.

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• B4.9

  Algebraic Curves

  Part II, 2003

  Let $X$ be a smooth curve of genus 0 over an algebraically closed field $k$. Show that $k(X)=k\left(\mathbb{P}^{1}\right) .$

  Now let $C$ be a plane projective curve defined by an irreducible homogeneous cubic polynomial.

  (a) Show that if $C$ is smooth then $C$ is not isomorphic to $\mathbb{P}^{1}$. Standard results on the canonical class may be assumed without proof, provided these are clearly stated.

  (b) Show that if $C$ has a singularity then there exists a non-constant morphism from $\mathbb{P}^{1}$ to $C$.

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• B1.13

  Probability and Measure

  Part II, 2003

  State and prove the first Borel-Cantelli Lemma.

  Suppose that $\left(F_{n}\right)$ is a sequence of events in a common probability space such that $\mathbb{P}\left(F_{i} \cap F_{j}\right) \leqslant \mathbb{P}\left(F_{i}\right) \cdot \mathbb{P}\left(F_{j}\right)$ whenever $i \neq j$ and that $\sum_{n} \mathbb{P}\left(F_{n}\right)=\infty$.

  Let $1_{F_{n}}$ be the indicator function of $F_{n}$ and let

  $$S_{n}=\sum_{k \leqslant n} 1_{F_{k}} ; \mu_{n}=\mathbb{E}\left(S_{n}\right)$$

  Use Chebyshev's inequality to show that

  $$\mathbb{P}\left(S_{n}<\frac{1}{2} \mu_{n}\right) \leqslant \mathbb{P}\left(\left|S_{n}-\mu_{n}\right|>\frac{1}{2} \mu_{n}\right) \leqslant \frac{4}{\mu_{n}}$$

  Deduce, using the first Borel-Cantelli Lemma, that $\mathbb{P}\left(F_{n}\right.$ infinitely often $)=1$.

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• B2.12

  Probability and Measure

  Part II, 2003

  Let $H$ be a Hilbert space and let $V$ be a closed subspace of $H$. Let $x \in H$. Show that there is a unique decomposition $x=u+v$ such that $v \in V$ and $u \in V^{\perp}$.

  Now suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is a probability space and let $X \in L^{2}(\Omega, \mathcal{F}, \mathbb{P})$. Suppose $\mathcal{G}$ is a sub- $\sigma$-algebra of $\mathcal{F}$. Define $\mathbb{E}(X \mid \mathcal{G})$ using a decomposition of the above type. Show that $\mathbb{E}\left(\mathbb{E}(X \mid \mathcal{G}) .1_{A}\right)=\mathbb{E}\left(X .1_{A}\right)$ for each set $A \in \mathcal{G}$.

  Let $\mathcal{G}_{1} \subseteq \mathcal{G}_{2}$ be two sub- $\sigma$-algebras of $\mathcal{F}$. Show that (a) $\mathbb{E}\left(\mathbb{E}\left(X \mid \mathcal{G}_{1}\right) \mid \mathcal{G}_{2}\right)=\mathbb{E}\left(X \mid \mathcal{G}_{1}\right)$; (b) $\mathbb{E}\left(\mathbb{E}\left(X \mid \mathcal{G}_{2}\right) \mid \mathcal{G}_{1}\right)=\mathbb{E}\left(X \mid \mathcal{G}_{1}\right)$.

  No general theorems about projections on Hilbert spaces may be quoted without proof.

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• B3.12

  Probability and Measure

  Part II, 2003

  Explain what is meant by the characteristic function $\phi$ of a real-valued random variable and prove that $|\phi|^{2}$ is also a characteristic function of some random variable.

  Let us say that a characteristic function $\phi$ is infinitely divisible when, for each $n \geqslant 1$, we can write $\phi=\left(\phi_{n}\right)^{n}$ for some characteristic function $\phi_{n}$. Prove that, in this case, the limit

  $$\psi(t)=\lim _{n \rightarrow \infty}\left|\phi_{2 n}(t)\right|^{2}$$

  exists for all real $t$ and is continuous at $t=0$.

  Using Lévy's continuity theorem for characteristic functions, which you should state carefully, deduce that $\psi$ is a characteristic function. Hence show that, if $\phi$ is infinitely divisible, then $\phi(t)$ cannot vanish for any real $t$.

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• B4.11

  Probability and Measure

  Part II, 2003

  Let $f:[a, b] \rightarrow \mathbb{R}$ be integrable with respect to Lebesgue measure $\mu$ on $[a, b]$. Prove that, if

  $$\int_{J} f d \mu=0$$

  for every sub-interval $J$ of $[a, b]$, then $f=0$ almost everywhere on $[a, b]$.

  Now define

  $$F(x)=\int_{a}^{x} f d \mu .$$

  Prove that $F$ is continuous on $[a, b]$. Show that, if $F$ is zero on $[a, b]$, then $f$ is zero almost everywhere on $[a, b]$.

  Suppose now that $f$ is bounded and Lebesgue integrable on $[a, b]$. By applying the Dominated Convergence Theorem to

  $$F_{n}(x)=\frac{F\left(x+\frac{1}{n}\right)-F(x)}{\frac{1}{n}},$$

  or otherwise, show that, if $F$ is differentiable on $[a, b]$, then $F^{\prime}=f$ almost everywhere on $[a, b]$.

  The functions $f_{n}:[a, b] \rightarrow \mathbb{R}$ have the properties:

  (a) $f_{n}$ converges pointwise to a differentiable function $g$ on $[a, b]$,

  (b) each $f_{n}$ has a continuous derivative $f_{n}^{\prime}$ with $\left|f_{n}^{\prime}(x)\right| \leqslant 1$ on $[a, b]$,

  (c) $f_{n}^{\prime}$ converges pointwise to some function $h$ on $[a, b]$.

  Deduce that

  $$h(x)=\lim _{n \rightarrow \infty}\left(\frac{d f_{n}(x)}{d x}\right)=\frac{d}{d x}\left(\lim _{n \rightarrow \infty} f_{n}(x)\right)=g^{\prime}(x)$$

  almost everywhere on $[a, b]$.

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• B2.13

  Applied Probability

  Part II, 2003

  Let $S_{k}$ be the sum of $k$ independent exponential random variables of rate $k \mu$. Compute the moment generating function of $S_{k}$.

  Consider, for each fixed $k$ and for $0<\lambda<\mu$, an $M / G / 1$ queue with arrival rate $\lambda$ and with service times distributed as $S_{k}$. Assume that the queue is empty at time 0 and write $T_{k}$ for the earliest time at which a customer departs leaving the queue empty. Show that, as $k \rightarrow \infty, T_{k}$ converges in distribution to a random variable $T$ whose moment generating function $M_{T}(\theta)$ satisfies

  $$\log \left(1-\frac{\theta}{\lambda}\right)+\log M_{T}(\theta)=\left(\frac{\theta-\lambda}{\mu}\right)\left(1-M_{T}(\theta)\right)$$

  Hence obtain the mean value of $T$.

  For what service-time distribution would the empty-to-empty time correspond exactly to $T$ ?

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• B3.13

  Applied Probability

  Part II, 2003

  State the product theorem for Poisson random measures.

  Consider a system of $n$ queues, each with infinitely many servers, in which, for $i=1, \ldots, n-1$, customers leaving the $i$ th queue immediately arrive at the $(i+1)$ th queue. Arrivals to the first queue form a Poisson process of rate $\lambda$. Service times at the $i$ th queue are all independent with distribution $F$, and independent of service times at other queues, for all $i$. Assume that initially the system is empty and write $V_{i}(t)$ for the number of customers at queue $i$ at time $t \geqslant 0$. Show that $V_{1}(t), \ldots, V_{n}(t)$ are independent Poisson random variables.

  In the case $F(t)=1-e^{-\mu t}$ show that

  $$\mathbb{E}\left(V_{i}(t)\right)=\frac{\lambda}{\mu} \mathbb{P}\left(N_{t} \geqslant i\right), \quad t \geqslant 0, \quad i=1, \ldots, n,$$

  where $\left(N_{t}\right)_{t \geqslant 0}$ is a Poisson process of rate $\mu$.

  Suppose now that arrivals to the first queue stop at time $T$. Determine the mean number of customers at the $i$ th queue at each time $t \geqslant T$.

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• B4.12

  Applied Probability

  Part II, 2003

  Explain what is meant by a renewal process and by a renewal-reward process.

  State and prove the law of large numbers for renewal-reward processes.

  A component used in a manufacturing process has a maximum lifetime of 2 years and is equally likely to fail at any time during that period. If the component fails whilst in use, it is replaced immediately by a similar component, at a cost of $£ 1000$. The factory owner may alternatively replace the component before failure, at a time of his choosing, at a cost of $£ 200$. What should the factory owner do?

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• B1.14

  Information Theory

  Part II, 2003

  A binary Huffman code is used for encoding symbols $1, \ldots, m$ occurring with probabilities $p_{1} \geqslant \ldots \geqslant p_{m}>0$ where $\sum_{1 \leqslant j \leqslant m} p_{j}=1$. Let $s_{1}$ be the length of a shortest codeword and $s_{m}$ of a longest codeword. Determine the maximal and minimal values of $s_{1}$ and $s_{m}$, and find binary trees for which they are attained.

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• B2.14

  Information Theory

  Part II, 2003

  Let $\mathcal{X}$ be a binary linear code of length $n$, rank $k$ and distance $d$. Let $x=$ $\left(x_{1}, \ldots, x_{n}\right) \in \mathcal{X}$ be a codeword with exactly $d$ non-zero digits.

  (a) Prove that $n \geqslant d+k-1$ (the Singleton bound).

  (b) Prove that truncating $\mathcal{X}$ on the non-zero digits of $x$ produces a code $\mathcal{X}^{\prime}$ of length $n-d$, rank $k-1$ and distance $d^{\prime}$ for some $d^{\prime} \geqslant\left\lceil\frac{d}{2}\right\rceil$. Here $\lceil a\rceil$ is the integer satisfying $a \leqslant\lceil a\rceil<a+1, a \in \mathbb{R}$.

  [Hint: Assume the opposite. Then, given $y \in \mathcal{X}$ and its truncation $y^{\prime} \in \mathcal{X}^{\prime}$, consider the coordinates where $x$ and $y$ have 1 in common (i.e. $x_{j}=y_{j}=1$ ) and where they differ $\left(\right.$ e.g. $x_{j}=1$ and $\left.y_{j}=0\right)$.]

  (c) Deduce that $n \geqslant d+\sum_{1 \leqslant \ell \leqslant k-1}\left\lceil\frac{d}{2^{\ell}}\right\rceil$ (an improved Singleton bound).

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• B4.13

  Information Theory

  Part II, 2003

  State and prove the Fano and generalized Fano inequalities.

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• B2.15

  Optimization and Control

  Part II, 2003

  The owner of a put option may exercise it on any one of the days $1, \ldots, h$, or not at all. If he exercises it on day $t$, when the share price is $x_{t}$, his profit will be $p-x_{t}$. Suppose the share price obeys $x_{t+1}=x_{t}+\epsilon_{t}$, where $\epsilon_{1}, \epsilon_{2}, \ldots$ are i.i.d. random variables for which $E\left|\epsilon_{t}\right|<\infty$. Let $F_{s}(x)$ be the maximal expected profit the owner can obtain when there are $s$ further days to go and the share price is $x$. Show that

  (a) $F_{s}(x)$ is non-decreasing in $s$,

  (b) $F_{s}(x)+x$ is non-decreasing in $x$, and

  (c) $F_{s}(x)$ is continuous in $x$.

  Deduce that there exists a non-decreasing sequence, $a_{1}, \ldots, a_{h}$, such that expected profit is maximized by exercising the option the first day that $x_{t} \leqslant a_{t}$.

  Now suppose that the option never expires, so effectively $h=\infty$. Show by examples that there may or may not exist an optimal policy of the form 'exercise the option the first day that $x_{t} \leqslant a$.

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• B3.14

  Optimization and Control

  Part II, 2003

  State Pontryagin's Maximum Principle (PMP).

  In a given lake the tonnage of fish, $x$, obeys

  $$d x / d t=0.001(50-x) x-u, \quad 0<x \leqslant 50$$

  where $u$ is the rate at which fish are extracted. It is desired to maximize

  $$\int_{0}^{\infty} u(t) e^{-0.03 t} d t$$

  choosing $u(t)$ under the constraints $0 \leqslant u(t) \leqslant 1.4$, and $u(t)=0$ if $x(t)=0$. Assume the PMP with an appropriate Hamiltonian $H(x, u, t, \lambda)$. Now define $G(x, u, t, \eta)=$ $e^{0.03 t} H(x, u, t, \lambda)$ and $\eta(t)=e^{0.03 t} \lambda(t)$. Show that there exists $\eta(t), 0 \leqslant t$ such that on the optimal trajectory $u$ maximizes

  $$G(x, u, t, \eta)=\eta[0.001(50-x) x-u]+u$$

  and

  $$d \eta / d t=0.002(x-10) \eta$$

  Suppose that $x(0)=20$ and that under an optimal policy it is not optimal to extract all the fish. Argue that $\eta(0) \geqslant 1$ is impossible and describe qualitatively what must happen under the optimal policy.

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• B4.14

  Optimization and Control

  Part II, 2003

  The scalars $x_{t}, y_{t}, u_{t}$, are related by the equations

  $$x_{t}=x_{t-1}+u_{t-1}, \quad y_{t}=x_{t-1}+\eta_{t-1}, \quad t=1, \ldots, T,$$

  where $\left\{\eta_{t}\right\}$ is a sequence of uncorrelated random variables with means of 0 and variances of 1. Given that $\hat{x}_{0}$ is an unbiased estimate of $x_{0}$ of variance 1 , the control variable $u_{t}$ is to be chosen at time $t$ on the basis of the information $W_{t}$, where $W_{0}=\left(\hat{x}_{0}\right)$ and $W_{t}=\left(\hat{x}_{0}, u_{0}, \ldots, u_{t-1}, y_{1}, \ldots, y_{t}\right), t=1,2, \ldots, T-1$. Let $\hat{x}_{1}, \ldots, \hat{x}_{T}$ be the Kalman filter estimates of $x_{1}, \ldots, x_{T}$ computed from

  $$\hat{x}_{t}=\hat{x}_{t-1}+u_{t-1}+h_{t}\left(y_{t}-\hat{x}_{t-1}\right)$$

  by appropriate choices of $h_{1}, \ldots, h_{T}$. Show that the variance of $\hat{x}_{t}$ is $V_{t}=1 /(1+t)$.

  Define $F\left(W_{T}\right)=E\left[x_{T}^{2} \mid W_{T}\right]$ and

  $$F\left(W_{t}\right)=\inf _{u_{t}, \ldots, u_{T-1}} E\left[\sum_{\tau=t}^{T-1} u_{\tau}^{2}+x_{T}^{2} \mid W_{t}\right], \quad t=0, \ldots, T-1$$

  Show that $F\left(W_{t}\right)=\hat{x}_{t}^{2} P_{t}+d_{t}$, where $P_{t}=1 /(T-t+1), d_{T}=1 /(1+T)$ and $d_{t-1}=V_{t-1} V_{t} P_{t}+d_{t} .$

  How would the expression for $F\left(W_{0}\right)$ differ if $\hat{x}_{0}$ had a variance different from $1 ?$

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• B1.17

  Dynamical Systems

  Part II, 2003

  Consider the one-dimensional map $f: \mathbb{R} \rightarrow \mathbb{R}$, where $f(x)=\mu x^{2}(1-x)$ with $\mu$ a real parameter. Find the range of values of $\mu$ for which the open interval $(0,1)$ is mapped into itself and contains at least one fixed point. Describe the bifurcation at $\mu=4$ and find the parameter value for which there is a period-doubling bifurcation. Determine whether the fixed point is an attractor at this bifurcation point.

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• B3.17

  Dynamical Systems

  Part II, 2003

  Let $f: I \rightarrow I$ be a continuous one-dimensional map of the interval $I \subset \mathbb{R}$. Explain what is meant by saying (a) that the map $f$ is topologically transitive, and (b) that the map $f$ has a horseshoe.

  Consider the tent map defined on the interval $[0,1]$ by

  $$f(x)= \begin{cases}\mu x & 0 \leqslant x<\frac{1}{2} \\ \mu(1-x) & \frac{1}{2} \leqslant x \leqslant 1\end{cases}$$

  for $1<\mu \leqslant 2$. Show that if $\mu>\sqrt{2}$ then this map is topologically transitive, and also that $f^{2}$ has a horseshoe.

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• B4.17

  Dynamical Systems

  Part II, 2003

  Let $f: S^{1} \rightarrow S^{1}$ be an orientation-preserving invertible map of the circle onto itself, with a lift $F: \mathbb{R} \rightarrow \mathbb{R}$. Define the rotation numbers $\rho_{0}(F)$ and $\rho(f)$.

  Suppose that $\rho_{0}(F)=p / q$, where $p$ and $q$ are coprime integers. Prove that the map $f$ has periodic points of least period $q$, and no periodic points with any least period not equal to $q$.

  Now suppose that $\rho_{0}(F)$ is irrational. Explain the distinction between wandering and non-wandering points under $f$. Let $\Omega(x)$ be the set of limit points of the sequence $\left\{x, f(x), f^{2}(x), \ldots\right\}$. Prove

  (a) that the set $\Omega(x)=\Omega$ is independent of $x$ and is the smallest closed, non-empty, $f$-invariant subset of $S^{1}$;

  (b) that $\Omega$ is the set of non-wandering points of $S^{1}$;

  (c) that $\Omega$ is either the whole of $S^{1}$ or a Cantor set in $S^{1}$.

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• B1.18

  Partial Differential Equations

  Part II, 2003

  (a) Define characteristic hypersurfaces and state a local existence and uniqueness theorem for a quasilinear partial differential equation with data on a non-characteristic hypersurface.

  (b) Consider the initial value problem

  $$3 u_{x}+u_{y}=-y u, \quad u(x, 0)=f(x),$$

  for a function $u: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $C^{1}$ initial data $f$ given for $y=0$. Obtain a formula for the solution by the method of characteristics and deduce that a $C^{1}$ solution exists for all $(x, y) \in \mathbb{R}^{2}$.

  Derive the following (well-posedness) property for solutions $u(x, y)$ and $v(x, y)$ corresponding to data $u(x, 0)=f(x)$ and $v(x, 0)=g(x)$ respectively:

  $$\sup _{x}|u(x, y)-v(x, y)| \leqslant \sup _{x}|f(x)-g(x)| \quad \text { for all } y .$$

  (c) Consider the initial value problem

  $$3 u_{x}+u_{y}=u^{2}, \quad u(x, 0)=f(x),$$

  for a function $u: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $C^{1}$ initial data $f$ given for $y=0$. Obtain a formula for the solution by the method of characteristics and hence show that if $f(x)<0$ for all $x$, then the solution exists for all $y>0$. Show also that if there exists $x_{0}$ with $f\left(x_{0}\right)>0$, then the solution does not exist for all $y>0$.

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• B2.17

  Partial Differential Equations

  Part II, 2003

  (a) If $f$ is a radial function on $\mathbb{R}^{n}$ (i.e. $f(x)=\phi(r)$ with $r=|x|$ for $x \in \mathbb{R}^{n}$ ), and $n>2$, then show that $f$ is harmonic on $\mathbb{R}^{n}-\{0\}$ if and only if

  $$\phi(r)=a+b r^{2-n}$$

  for $a, b \in \mathbb{R}$.

  (b) State the mean value theorem for harmonic functions and prove it for $n>2$.

  (c) Generalise the statement and the proof of the mean value theorem to the case of a subharmonic function, i.e. a $C^{2}$ function such that $\Delta u \leqslant 0$.

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• B3.18

  Partial Differential Equations

  Part II, 2003

  Consider the initial value problem

  $$\frac{\partial^{2} u}{\partial t^{2}}-\Delta u=0$$

  to be solved for $u: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$, subject to the initial conditions

  $$u(0, x)=f(x), \quad \frac{\partial u}{\partial t}(0, x)=0$$

  for $f$ in the Schwarz space $\mathcal{S}\left(\mathbb{R}^{n}\right)$. Use the Fourier transform in $x$ to obtain a representation for the solution in the form

  $$u(t, x)=\int e^{i x \cdot \xi} A(t, \xi) \widehat{f}(\xi) d^{n} \xi$$

  where $A$ should be determined explicitly. Explain carefully why your formula gives a smooth solution to (1) and why it satisfies the initial conditions (2), referring to the required properties of the Fourier transform as necessary.

  Next consider the case $n=1$. Find a tempered distribution $T$ (depending on $t, x$ ) such that (3) can be written

  $$u=<T, \widehat{f}>$$

  and (using the definition of Fourier transform of tempered distributions) show that the formula reduces to

  $$u(t, x)=\frac{1}{2}[f(x-t)+f(x+t)]$$

  State and prove the Duhamel principle relating to the solution of the $n$-dimensional inhomogeneous wave equation

  $$\frac{\partial^{2} u}{\partial t^{2}}-\Delta u=h$$

  to be solved for $u: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$, subject to the initial conditions

  $$u(0, x)=0, \quad \frac{\partial u}{\partial t}(0, x)=0$$

  for $h$ a $C^{\infty}$ function. State clearly assumptions used on the solvability of the homogeneous problem.

  [Hint: it may be useful to consider the Fourier transform of the tempered distribution defined by the function $\xi \mapsto e^{i \xi \cdot a}$.]

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• B4.18

  Partial Differential Equations

  Part II, 2003

  Discuss the basic properties of the Fourier transform and how it is used in the study of partial differential equations.

  The essay should include: definition and basic properties, inversion theorem, applications to establishing well-posedness of evolution partial differential equations with constant coefficients.

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• B1.19

  Methods of Mathematical Physics

  Part II, 2003

  By considering the integral

  $$\int_{C}\left(\frac{t}{1-t}\right)^{i} d t$$

  where $C$ is a large circle centred on the origin, show that

  $$B(1+i, 1-i)=\pi \operatorname{cosech} \pi$$

  where

  $$B(p, q)=\int_{0}^{1} t^{p-1}(1-t)^{q-1} d t, \operatorname{Re}(p)>0, \operatorname{Re}(q)>0$$

  By using $B(p, q)=\frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)}$, deduce that $\Gamma(i) \Gamma(-i)=\pi \operatorname{cosech} \pi$.

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• B2.18

  Methods of Mathematical Physics

  Part II, 2003

  Let $\widehat{y}(p)$ be the Laplace transform of $y(t)$, where $y(t)$ satisfies

  $$y^{\prime}(t)=y(\pi-t)$$

  and

  $$y(0)=1 ; \quad y(\pi)=k ; \quad y(t)=0 \text { for } t<0 \text { and for } t>\pi$$

  Show that

  $$p \widehat{y}(p)+k e^{-\pi p}-1=e^{-\pi p} \widehat{y}(-p)$$

  and hence deduce that

  $$\widehat{y}(p)=\frac{(k+p)-(1+p k) e^{-\pi p}}{1+p^{2}}$$

  Use the inversion formula for Laplace transforms to find $y(t)$ for $t>\pi$ and deduce that a solution of the above boundary value problem exists only if $k=-1$. Hence find $y(t)$ for $0 \leqslant t \leqslant \pi$.

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• B3.19

  Methods of Mathematical Physics

  Part II, 2003

  Let

  $$f(\lambda)=\int_{\gamma} e^{\lambda\left(t-t^{3} / 3\right)} d t, \quad \lambda \text { real and positive }$$

  where $\gamma$ is a path beginning at $\infty e^{-2 i \pi / 3}$ and ending at $+\infty$ (on the real axis). Identify the saddle points and sketch the paths of constant phase through these points.

  Hence show that $f(\lambda) \sim e^{2 \lambda / 3} \sqrt{\pi / \lambda}$ as $\lambda \rightarrow \infty$.

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• B4.19

  Methods of Mathematical Physics

  Part II, 2003

  By setting $w(z)=\int_{\gamma} f(t) e^{-z t} d t$, where $\gamma$ and $f(t)$ are to be suitably chosen, explain how to find integral representations of the solutions of the equation

  $$z w^{\prime \prime}-k w=0$$

  where $k$ is a non-zero real constant and $z$ is complex. Discuss $\gamma$ in the particular case that $z$ is restricted to be real and positive and distinguish the different cases that arise according to the $\operatorname{sign}$ of $k$.

  Show that in this particular case, by choosing $\gamma$ as a closed contour around the origin, it is possible to express a solution in the form

  $$w(z)=A \sum_{n=0}^{\infty} \frac{(z k)^{n+1}}{n !(n+1) !}$$

  where $A$ is a constant.

  Show also that for $k>0$ there are solutions that satisfy

  $$w(z) \sim B z^{1 / 4} e^{-2 \sqrt{k z}} \quad \text { as } z \rightarrow \infty$$

  where $B$ is a constant.

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• B1.21

  Electrodynamics

  Part II, 2003

  A particle of charge $q$ and mass $m$ moves non-relativistically with 4 -velocity $u^{a}(t)$ along a trajectory $x^{a}(t)$. Its electromagnetic field is determined by the Liénard-Wiechert potential

  $$A^{a}\left(\mathbf{x}^{\prime}, t^{\prime}\right)=\frac{q}{4 \pi \epsilon_{0}} \frac{u^{a}(t)}{u_{b}(t)\left(x^{\prime}-x(t)\right)^{b}}$$

  where $t^{\prime}=t+\left|\mathbf{x}-\mathbf{x}^{\prime}\right|$ and $\mathbf{x}$ denotes the spatial part of the 4 -vector $x^{a}$.

  Derive a formula for the Poynting vector at very large distances from the particle. Hence deduce Larmor's formula for the rate of loss of energy due to electromagnetic radiation by the particle.

  A particle moves in the $(x, y)$ plane in a constant magnetic field $\mathbf{B}=(0,0, B)$. Initially it has kinetic energy $E_{0}$; derive a formula for the kinetic energy of this particle as a function of time.

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• B2.20

  Electrodynamics

  Part II, 2003

  A plane electromagnetic wave of frequency $\omega$ and wavevector $\mathbf{k}$ has an electromagnetic potential given by

  $$A^{a}=A \epsilon^{a} e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}$$

  where $A$ is the amplitude of the wave and $\epsilon^{a}$ is the polarization vector. Explain carefully why there are two independent polarization states for such a wave, and why $|\mathbf{k}|^{2}=\omega^{2}$.

  A wave travels in the positive $z$-direction with polarization vector $\epsilon^{a}=(0,1, i, 0)$. It is incident at $z=0$ on a plane surface which conducts perfectly in the $x$-direction, but not at all in the $y$-direction. Find an expression for the electromagnetic potential of the radiation that is reflected from this surface.

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• B4.21

  Electrodynamics

  Part II, 2003

  Describe the physical meaning of the various components of the stress-energy tensor $T^{a b}$ of the electromagnetic field.

  Suppose that one is given an electric field $\mathbf{E}(\mathbf{x})$ and a magnetic field $\mathbf{B}(\mathbf{x})$. Show that the angular momentum about the origin of these fields is

  $$\mathbf{J}=\frac{1}{\mu_{0}} \int \mathbf{x} \times(\mathbf{E} \times \mathbf{B}) d^{3} \mathbf{x}$$

  where the integral is taken over all space.

  A point electric charge $Q$ is at the origin, and has electric field

  $$\mathbf{E}=\frac{Q}{4 \pi \epsilon_{0}} \frac{\mathbf{x}}{|\mathbf{x}|^{3}}$$

  A point magnetic monopole of strength $P$ is at $\mathbf{y}$ and has magnetic field

  $$\mathbf{B}=\frac{\mu_{0} P}{4 \pi} \frac{\mathbf{x}-\mathbf{y}}{|\mathbf{x}-\mathbf{y}|^{3}}$$

  Find the component, along the axis between the electric charge and the magnetic monopole, of the angular momentum of the electromagnetic field about the origin.

  [Hint: You may find it helpful to express both $\mathbf{E}$ and $\mathbf{B}$ as gradients of scalar potentials.]

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• B1.22

  Statistical Physics

  Part II, 2003

  A gas in equilibrium at temperature $T$ and pressure $P$ has quantum stationary states $i$ with energies $E_{i}(V)$ in volume $V$. What does it mean to say that a change in volume from $V$ to $V+d V$ is reversible?

  Write down an expression for the probability that the gas is in state $i$. How is the entropy $S$ defined in terms of these probabilities? Write down an expression for the energy $E$ of the gas, and establish the relation

  $$d E=T d S-P d V$$

  for reversible changes.

  By considering the quantity $F=E-T S$, derive the Maxwell relation

  $$\left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial P}{\partial T}\right)_{V}$$

  A gas obeys the equation of state

  $$P V=R T+\frac{B(T)}{V}$$

  where $R$ is a constant and $B(T)$ is a function of $T$ only. The gas is expanded isothermally, at temperature $T$, from volume $V_{0}$ to volume $2 V_{0}$. Find the work $\Delta W$ done on the gas. Show that the heat $\Delta Q$ absorbed by the gas is given by

  $$\Delta Q=R T \log 2+\frac{T}{2 V_{0}} \frac{d B}{d T}$$

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• B3.22

  Statistical Physics

  Part II, 2003

  A diatomic molecule, free to move in two space dimensions, has classical Hamiltonian

  $$H=\frac{1}{2 m}|\mathbf{p}|^{2}+\frac{1}{2 I} J^{2}$$

  where $\mathbf{p}=\left(p_{1}, p_{2}\right)$ is the particle's momentum and $J$ is its angular momentum. Write down the classical partition function $Z$ for an ideal gas of $N$ such molecules in thermal equilibrium at temperature $T$. Show that it can be written in the form

  $$Z=\left(z_{t} z_{r o t}\right)^{N}$$

  where $z_{t}$ and $z_{\text {rot }}$ are the one-molecule partition functions associated with the translational and rotational degrees of freedom, respectively. Compute $z_{t}$ and $z_{r o t}$ and hence show that the energy $E$ of the gas is given by

  $$E=\frac{3}{2} N k T$$

  where $k$ is Boltzmann's constant. How does this result illustrate the principle of equipartition of energy?

  In an improved model of the two-dimensional gas of diatomic molecules, the angular momentum $J$ is quantized in integer multiples of $\hbar$ :

  $$J=j \hbar, \quad j=0, \pm 1, \pm 2, \ldots$$

  Write down an expression for $z_{\text {rot }}$ in this case. Given that $k T \ll\left(\hbar^{2} / 2 I\right)$, obtain an expression for the energy $E$ in the form

  $$E \approx A T+B e^{-\hbar^{2} / 2 I k T}$$

  where $A$ and $B$ are constants that should be computed. How is this result compatible with the principle of equipartition of energy? Find $C_{v}$, the specific heat at constant volume, for $k T \ll\left(\hbar^{2} / 2 I\right)$.

  Why can the sum over $j$ in $z_{\text {rot }}$ be approximated by an integral when $k T \gg\left(\hbar^{2} / 2 I\right)$ ? Deduce that $E \approx \frac{3}{2} N k T$ in this limit.

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• B4.23

  Statistical Physics

  Part II, 2003

  A gas of non-interacting identical bosons in volume $V$, with one-particle energy levels $\epsilon_{r}, r=1,2, \ldots, \infty$, is in equilibrium at temperature $T$ and chemical potential $\mu$. Let $n_{r}$ be the number of particles in the $r$ th one-particle state. Write down an expression for the grand partition function $\mathcal{Z}$. Write down an expression for the probability of finding a given set of occupation numbers $n_{r}$ of the one-particle states. Hence determine the mean occupation numbers $\bar{n}_{r}$ (the Bose-Einstein distribution). Write down expressions, in terms of the mean occupation numbers, for the total energy $E$ and total number of particles $N$.

  Write down an expression for the grand potential $\Omega$ in terms of $\mathcal{Z}$. Given that

  $$S=-\left(\frac{\partial \Omega}{\partial T}\right)_{V, \mu}$$

  show that $S$ can be written in the form

  $$S=k \sum_{r} f\left(\bar{n}_{r}\right)$$

  for some function $f$, which you should determine. Hence show that $d S=0$ for any change of the gas that leaves the mean occupation numbers unchanged. Consider a (quasi-static) change of $V$ with this property. Using the formula

  $$P=-\left(\frac{\partial E}{\partial V}\right)_{N, S}$$

  and given that $\epsilon_{r} \propto V^{-\sigma}(\sigma>0)$ for each $r$, show that

  $$P=\sigma(E / V)$$

  What is the value of $\sigma$ for photons?

  Let $\mu=0$, so that $E$ is a function only of $T$ and $V$. Why does the energy density $\varepsilon=E / V$ depend only on $T ?$ Using the Maxwell relation

  $$\left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial P}{\partial T}\right)_{V}$$

  and the first law of thermodynamics for reversible changes, show that

  $$\left(\frac{\partial E}{\partial V}\right)_{T}=T\left(\frac{\partial P}{\partial T}\right)_{V}-P$$

  and hence that

  $$\varepsilon(T) \propto T^{\gamma}$$

  for some power $\gamma$ that you should determine. Show further that

  $$S \propto\left(T V^{\sigma}\right)^{\frac{1}{\sigma}} .$$

  Hence verify, given $\mu=0$, that $\bar{n}_{r}$ is left unchanged by a change of $V$ at constant $S$.

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• B1.23

  Applications of Quantum Mechanics

  Part II, 2003

  Define the differential cross section $\frac{d \sigma}{d \Omega}$. Show how it may be related to a scattering amplitude $f$ defined in terms of the behaviour of a wave function $\psi$ satisfying suitable boundary conditions as $r=|\mathbf{r}| \rightarrow \infty$.

  For a particle scattering off a potential $V(r)$ show how $f(\theta)$, where $\theta$ is the scattering angle, may be expanded, for energy $E=\hbar^{2} k^{2} / 2 m$, as

  $$f(\theta)=\sum_{\ell=0}^{\infty} f_{\ell}(k) P_{\ell}(\cos \theta),$$

  and find $f_{\ell}(k)$ in terms of the phase shift $\delta_{\ell}(k)$. Obtain the optical theorem relating $\sigma_{\text {total }}$ and $f(0)$.

  Suppose that

  $$e^{2 i \delta_{1}}=\frac{E-E_{0}-\frac{1}{2} i \Gamma}{E-E_{0}+\frac{1}{2} i \Gamma}$$

  Why for $E \approx E_{0}$ may $f_{1}(k)$ be dominant, and what is the expected behaviour of $\frac{d \sigma}{d \Omega}$ for $E \approx E_{0}$ ?

  [For large $r$

  $$e^{i k r \cos \theta} \sim \frac{1}{2 i k r} \sum_{\ell=0}^{\infty}(2 \ell+1)\left((-1)^{\ell+1} e^{-i k r}+e^{i k r}\right) P_{\ell}(\cos \theta)$$

  Legendre polynomials satisfy

  $$\left.\int_{-1}^{1} P_{\ell}(t) P_{\ell^{\prime}}(t) d t=\frac{2}{2 \ell+1} \delta_{\ell \ell^{\prime}} \cdot\right]$$

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• B2.22

  Applications of Quantum Mechanics

  Part II, 2003

  The Hamiltonian $H_{0}$ for a single electron atom has energy eigenstates $\left|\psi_{n}\right\rangle$ with energy eigenvalues $E_{n}$. There is an interaction with an electromagnetic wave of the form

  $$H_{1}=-e \mathbf{r} \cdot \boldsymbol{\epsilon} \cos (\mathbf{k} \cdot \mathbf{r}-\omega t), \quad \omega=|\mathbf{k}| c,$$

  where $\boldsymbol{\epsilon}$ is the polarisation vector. At $t=0$ the atom is in the state $\left|\psi_{0}\right\rangle$. Find a formula for the probability amplitude, to first order in $e$, to find the atom in the state $\left|\psi_{1}\right\rangle$ at time $t$. If the atom has a size $a$ and $|\mathbf{k}| a \ll 1$ what are the selection rules which are relevant? For $t$ large, under what circumstances will the transition rate be approximately constant?

  [You may use the result

  $$\left.\int_{-\infty}^{\infty} \frac{\sin ^{2} \lambda t}{\lambda^{2}} d \lambda=\pi|t| . \quad\right]$$

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• B3.23

  Applications of Quantum Mechanics

  Part II, 2003

  Consider the two Hamiltonians

  $$\begin{aligned} &H_{1}=\frac{\mathbf{p}^{2}}{2 m}+V(|\mathbf{r}|), \\ &H_{2}=\frac{\mathbf{p}^{2}}{2 m}+\sum_{n_{i} \in \mathbb{Z}} V\left(\left|\mathbf{r}-n_{1} \mathbf{a}_{1}-n_{2} \mathbf{a}_{2}-n_{3} \mathbf{a}_{3}\right|\right), \end{aligned}$$

  where $\mathbf{a}_{i}$ are three linearly independent vectors. For each of the Hamiltonians $H=H_{1}$ and $H=H_{2}$, what are the symmetries of $H$ and what unitary operators $U$ are there such that $U H U^{-1}=H$ ?

  For $\mathrm{H}_{2}$ derive Bloch's theorem. Suppose that $H_{1}$ has energy eigenfunction $\psi_{0}(\mathbf{r})$ with energy $E_{0}$ where $\psi_{0}(\mathbf{r}) \sim N e^{-K r}$ for large $r=|\mathbf{r}|$. Assume that $K\left|\mathbf{a}_{i}\right| \gg 1$ for each $i$. In a suitable approximation derive the energy eigenvalues for $H_{2}$ when $E \approx E_{0}$. Verify that the energy eigenfunctions and energy eigenvalues satisfy Bloch's theorem. What happens if $K\left|\mathbf{a}_{i}\right| \rightarrow \infty$ ?

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• B4.24

  Applications of Quantum Mechanics

  Part II, 2003

  Atoms of mass $m$ in an infinite one-dimensional periodic array, with interatomic spacing $a$, have perturbed positions $x_{n}=n a+y_{n}$, for integer $n$. The potential between neighbouring atoms is

  $$\frac{1}{2} \lambda\left(x_{n+1}-x_{n}-a\right)^{2}$$

  for positive constant $\lambda$. Write down the Lagrangian for the variables $y_{n}$. Find the frequency $\omega(k)$ of a normal mode of wavenumber $k$. Define the Brillouin zone and explain why $k$ may be restricted to lie within it.

  Assume now that the array is periodically-identified, so that there are effectively only $N$ atoms in the array and the atomic displacements $y_{n}$ satisfy the periodic boundary conditions $y_{n+N}=y_{n}$. Determine the allowed values of $k$ within the Brillouin zone. Show, for allowed wavenumbers $k$ and $k^{\prime}$, that

  $$\sum_{n=0}^{N-1} e^{i n\left(k-k^{\prime}\right) a}=N \delta_{k, k^{\prime}}$$

  By writing $y_{n}$ as

  $$y_{n}=\frac{1}{\sqrt{N}} \sum_{k} q_{k} e^{i n k a}$$

  where the sum is over allowed values of $k$, find the Lagrangian for the variables $q_{k}$, and hence the Hamiltonian $H$ as a function of $q_{k}$ and the conjugate momenta $p_{k}$. Show that the Hamiltonian operator $\hat{H}$ of the quantum theory can be written in the form

  $$\hat{H}=E_{0}+\sum_{k} \hbar \omega(k) a_{k}^{\dagger} a_{k}$$

  where $E_{0}$ is a constant and $a_{k}, a_{k}^{\dagger}$ are harmonic oscillator annihilation and creation operators. What is the physical interpretation of $a_{k}$ and $a_{k}^{\dagger}$ ? How does this show that phonons have quantized energies?

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• B1.25

  Fluid Dynamics II

  Part II, 2003

  Consider a two-dimensional horizontal vortex sheet of strength $U$ at height $h$ above a horizontal rigid boundary at $y=0$, so that the inviscid fluid velocity is

  $$\mathbf{u}= \begin{cases}(U, 0) & 0<y<h \\ (0,0) & y>h\end{cases}$$

  Examine the temporal linear instabililty of the sheet and determine the relevant dispersion relationship.

  For what wavelengths is the sheet unstable?

  Evaluate the temporal growth rate and the wave propagation speed in the limit of both short and long waves. Comment briefly on the significance of your results.

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• B2 24

  Fluid Dynamics II

  Part II, 2003

  A plate is drawn vertically out of a bath and the resultant liquid drains off the plate as a thin film. Using lubrication theory, show that the governing equation for the thickness of the film, $h(x, t)$ is

  $$\frac{\partial h}{\partial t}+\left(\frac{g h^{2}}{\nu}\right) \frac{\partial h}{\partial x}=0$$

  where $t$ is time and $x$ is the distance down the plate measured from the top.

  Verify that

  $$h(x, t)=F\left(x-\frac{g h^{2}}{\nu} t\right)$$

  satisfies $(*)$ and identify the function $F(x)$. Using this relationship or otherwise, determine an explicit expression for the thickness of the film assuming that it was initially of uniform thickness $h_{0}$.

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• B3.24

  Fluid Dynamics II

  Part II, 2003

  A steady two-dimensional jet is generated in an infinite, incompressible fluid of density $\rho$ and kinematic viscosity $\nu$ by a point source of momentum with momentum flux in the $x$ direction $F$ per unit length located at the origin.

  Using boundary layer theory, analyse the motion in the jet and show that the $x$-component of the velocity is given by

  $$u=U(x) f^{\prime}(\eta)$$

  where

  $$\eta=y / \delta(x), \quad \delta(x)=\left(\rho \nu^{2} x^{2} / F\right)^{1 / 3} \text { and } U(x)=\left(F^{2} / \rho^{2} \nu x\right)^{1 / 3}$$

  Show that $f$ satisfies the differential equation

  $$f^{\prime \prime \prime}+\frac{1}{3}\left(f f^{\prime \prime}+f^{\prime^{2}}\right)=0 .$$

  Write down the appropriate boundary conditions for this equation. [You need not solve the equation.]

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• B4.26

  Fluid Dynamics II

  Part II, 2003

  Show that the complex potential in the complex $\zeta$ plane,

  $$w=(U-i V) \zeta+(U+i V) \frac{c^{2}}{\zeta}-\frac{i \kappa}{2 \pi} \log \zeta$$

  describes irrotational, inviscid flow past the rigid cylinder $|\zeta|=c$, placed in a uniform flow $(U, V)$ with circulation $\kappa$.

  Show that the transformation

  $$z=\zeta+\frac{c^{2}}{\zeta}$$

  maps the circle $|\zeta|=c$ in the $\zeta$ plane onto the flat plate airfoil $-2 c<x<2 c, y=0$ in the $z$ plane $(z=x+i y)$. Hence, write down an expression for the complex potential, $w_{p}$, of uniform flow past the flat plate, with circulation $\kappa$. Indicate very briefly how the value of $\kappa$ might be chosen to yield a physical solution.

  Calculate the turning moment, $M$, exerted on the flat plate by the flow.

  (You are given that

  $$M=-\frac{1}{2} \rho \operatorname{Re}\left\{\oint\left[\frac{\left(\frac{d w}{d \zeta}\right)^{2}}{\frac{d z}{d \zeta}}\right] z(\zeta) d \zeta\right\} \text {, }$$

  where $\rho$ is the fluid density and the integral is to be completed around a contour enclosing the circle $|\zeta|=c$ ).

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• B1.26

  Waves in Fluid and Solid Media

  Part II, 2003

  Consider the equation

  $$\frac{\partial^{2} \phi}{\partial t^{2}}+\alpha^{2} \frac{\partial^{4} \phi}{\partial x^{4}}+\beta^{2} \phi=0$$

  with $\alpha$ and $\beta$ real constants. Find the dispersion relation for waves of frequency $\omega$ and wavenumber $k$. Find the phase velocity $c(k)$ and the group velocity $c_{g}(k)$, and sketch the graphs of these functions.

  By multiplying $(*)$ by $\partial \phi / \partial t$, obtain an energy equation in the form

  $$\frac{\partial E}{\partial t}+\frac{\partial F}{\partial x}=0$$

  where $E$ represents the energy density and $F$ the energy flux.

  Now let $\phi(x, t)=A \cos (k x-\omega t)$, where $A$ is a real constant. Evaluate the average values of $E$ and $F$ over a period of the wave to show that

  $$\langle F\rangle=c_{g}\langle E\rangle$$

  Comment on the physical meaning of this result.

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• B2.25

  Waves in Fluid and Solid Media

  Part II, 2003

  Starting from the equations for the one-dimensional unsteady flow of a perfect gas of uniform entropy, derive the Riemann invariants

  $$u \pm \frac{2}{\gamma-1} c=\text { constant }$$

  on characteristics

  $$C_{\pm}: \frac{d x}{d t}=u \pm c$$

  A piston moves smoothly down a long tube, with position $x=X(t)$. Gas occupies the tube ahead of the piston, $x>X(t)$. Initially the gas and the piston are at rest, and the speed of sound in the gas is $c_{0}$. For $t>0$, show that the $C_{+}$characteristics are straight lines, provided that a shock-wave has not formed. Hence find a parametric representation of the solution for the velocity $u(x, t)$ of the gas.

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• B3.25

  Waves in Fluid and Solid Media

  Part II, 2003

  Derive the wave equation governing the velocity potential for linearised sound in a perfect gas. How is the pressure disturbance related to the velocity potential? Write down the spherically symmetric solution to the wave equation with time dependence $e^{i \omega t}$, which is regular at the origin.

  A high pressure gas is contained, at density $\rho_{0}$, within a thin metal spherical shell which makes small amplitude spherically symmetric vibrations. Ignore the low pressure gas outside. Let the metal shell have radius $a$, mass $m$ per unit surface area, and elastic stiffness which tries to restore the radius to its equilibrium value $a_{0}$ with a force $-\kappa\left(a-a_{0}\right)$ per unit surface area. Show that the frequency of these vibrations is given by

  $$\omega^{2}\left(m+\frac{\rho_{0} a_{0}}{\theta \cot \theta-1}\right)=\kappa \quad \text { where } \theta=\omega a_{0} / c_{0}$$

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• B4.27

  Waves in Fluid and Solid Media

  Part II, 2003

  Show that the equations governing isotropic linear elasticity have plane-wave solutions, identifying them as $\mathrm{P}, \mathrm{SV}$ or $\mathrm{SH}$ waves.

  A semi-infinite elastic medium in $y<0$ (where $y$ is the vertical coordinate) with density $\rho$ and Lamé moduli $\lambda$ and $\mu$ is overlaid by a layer of thickness $h$ (in $0<y<h$ ) of a second elastic medium with density $\rho^{\prime}$ and Lamé moduli $\lambda^{\prime}$ and $\mu^{\prime}$. The top surface at $y=h$ is free, i.e. the surface tractions vanish there. The speed of S-waves is lower in the layer, i.e. $c_{S}^{\prime}{ }^{2}=\mu^{\prime} / \rho^{\prime}<\mu / \rho=c_{S}^{2}$. For a time-harmonic SH-wave with horizontal wavenumber $k$ and frequency $\omega$, which oscillates in the slow top layer and decays exponentially into the fast semi-infinite medium, derive the dispersion relation for the apparent wave speed $c(k)=\omega / k$,

  $$\tan \left(k h \sqrt{\frac{c^{2}}{c_{S}^{2}}-1}\right)=\frac{\mu \sqrt{1-\frac{c^{2}}{c_{S}^{2}}}}{\mu^{\prime} \sqrt{\frac{c^{2}}{c_{S}^{\prime}}-1}} .$$

  Show graphically that there is always one root, and at least one higher mode if $\sqrt{c_{S}^{2} / c_{S}^{\prime 2}-1}>\pi / k h$.

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